
Glass I 

Book_ 



£6 



CopyiightN°_ 



COPYRIGHT DEPOSIT. 



L 



APPLIED THERMODYNAMICS 

FOR ENGINEERS 



BY 



WILLIAM D. ENNIS, M.E. 

MEMBER OF AMERICAN SOCIETY OF MECHANICAL ENGINEERS 

PROFESSOR OF MECHANICAL ENGINEERING IN THE 

POLYTECHNIC INSTITUTE OF BROOKLYN 



With 316 Illustrations 




NEW YORK 
D. VAN NOSTRAND COMPANY 

23 Murray and 27 Warren Streets 

1910 






^ 



Copyright, 1910, by 
D. Van Nostrand Company 



A I 



©CIA273020 



PREFACE 

" Applied Thermodynamics " is a pretty broad title ; but it is 
intended to describe a method of treatment rather than unusual 
scope. The writer's aim has been to present those fundamental 
principles which concern the designer no less than the technical 
student in such a way as to convince of their importance. 

The vital problem of the day in mechanical engineering is that 
of the prime mover. Is the steam engine, the gas engine, or the 
turbine to survive? The internal combustion engine works with 
the wide range of temperature shown by Carnot to be desirable ; 
but practically its superiority in efficiency is less marked than its 
temperature range should warrant. In most forms, its entire charge, 
and in all forms, the greater part of its charge, must be compressed 
by a separate and thermally wasteful operation. By using liquid 
or solid fuel, this complication may be limited so as to apply to the 
air supply only ; but as this air supply constitutes the greater part 
of the combustible mixture, the difficulties remain serious, and there 
is no present means available for supplying oxygen in liquid or solid 
form so as to wholly avoid the necessity for compression. 

The turbine, with superheat and high vacuum, has not yet 
surpassed the best efficiency records of the reciprocating engine, 
although commercially its superior in many applications. Like the 
internal combustion engine, the turbine, with its wide temperature 
range, has gone far toward offsetting its low efficiency ratio ; where 
the temperature range has been narrow the economy has been low, 
and when running non-condensing the efficiency of the turbine has 
compared unfavorably with that of the engine. There is promise 
of development along the line of attack on the energy losses in the 
turbine; there seems little to be accomplished in reducing these 
losses in the engine. The two motors may at any moment reach 
a parity. 

iii 



iv PREFACE 

These are the questions which should be kept in mind by the 
reader. Thermodynamics is physics, not mathematics or logic. 
This book takes a middle ground between those text-books which 
replace all theory by empiricism and that other class of treatises 
which are too apt to ignore the engineering significance of their 
vocabulary of differential equations. We here aim to present ideal 
operations, to show how they are modified in practice, to amplify 
underlying principles, and to stop when the^ further application of 
those principles becomes a matter of machine design. Thermo- 
dynamics has its own distinct and by no means narrow scope, and 
the intellectual training arising from its study is not to be ignored. 
We here deal only with a few of its engineering aspects ; but these, 
with all others, hark back invariably to a few fundamental princi- 
ples, and these principles are the matters for insistent emphasis. 
Too much anxiety is sometimes shown to quickly reach rules of 
practice. This, perhaps, has made our subject too often the barren 
science. Rules of practice eternally change ; for they depend not 
alone on underlying theory, but on conditions current. Our theory 
should be so sound, and our grasp of underlying principles so just, 
that we may successfully attack new problems as they arise and 
evolve those rules of practice which at any moment may be best 
for the conditions existing at that moment. 

But if Thermodynamics is not differential equations, neither 
should too much trouble be taken to avoid the use of mathematics 
which every engineer is supposed to have mastered. The calculus 
is accordingly employed where it saves time and trouble, not else- 
where. The so-called general mathematical method has been used 
in the one application where it is still necessary ; elsewhere, special 
methods, which give more physical significance to the things de- 
scribed, have been employed in preference. Formulas are useful 
to the busy engineer, but destructive to the student ; and after 
weighing the matter the writer has chosen to avoid formal definitions 
and too binding symbols, preferring to compel the occasionally 
reluctant reader to grub out roots for himself — an excellent exer- 
cise which becomes play by practice. 

The subject of compressed air is perhaps not Thermodynamics, 
but it illustrates in a simple way many of the principles of gases 



PREFACE v 

and has therefore been included. Some other topics may convey 
an impression of novelty ; the gas engine is treated before the steam 
engine, because if the order is reversed the reader will usually be 
rusty on the theory of gases after spending some weeks with vapor 
phenomena ; a brief exposition of multiple-effect distillation is pre- 
sented; a limit is suggested for the efficiency of the power gas 
producer ; and, carrying out the general use of the entropy diagram 
for illustrative purposes, new entropy charts have been prepared 
for ammonia, ether, and carbon dioxide. A large number of prob- 
lems has been incorporated. Most of these should be worked with 
the aid of the slide rule. 

Further originality is not claimed. The subject has been written, 
and may now be only re-presented. All standard works have been 
consulted, and an effort has been made to give credit for methods 
as well as data. Yet it would be impossible in this way to fully 
acknowledge the beneficial influence of the writer's former teachers, 
the late Professor Wood, Professor J. E. Denton, and Dr. D. S. 
Jacobus. It may be sufficient to say that if there is anything good 
in the book they have contributed to it ; and for what is not good, 
they are not responsible. 

Polytechnic Institute of Brooklyn, 
New York, August, 1910. 



CONTENTS 

CHAPTER PAGE 

I. The Nature and Effects of Heat 1 

II. The Heat Unit : Specific Heat : First Law of Thermo- 
dynamics 11 

III. Laws of Gases : Absolute Temperature : The Perfect Gas 19 

IV. Thermal Capacities : Specific Heats of Gases : Joule's Law 29 

V. Graphical Representations : Pressure-volume Paths of Per- 
fect Gases 39 

VI. The Carnot Cycle 63 

VII. The Second Law of Thermodynamics 70 

VIII. Entropy 77 

IX. Compressed Air .90 

The cold air engine.: cycle, temperature fall, preheaters, design of 
engine : the compressor : cycle, form of compression curve, 
jackets, multi-stage compression, intercooling, power consump- 
tion: engine and compressor relations: losses, efficiencies, en- 
tropy diagram, compressor capacity, volumetric efficiency, 
design of compressor, commercial types : compressed air trans- 
mission. 

X. Hot-air Engines 129 

XI. Gas Power 146 

The producer : limit of efficiency : gas engine cycles : Otto, Car- 
not, Atkinson, Lenoir, Bray ton, Clerk, Diesel: practical modi- 
fications of the Otto cycle: mixture, compression, ignition, 
dissociation, clearance, expansion, scavenging, diagram factor: 
principles of design and efficiency : commercial gas engines : 
results of tests : gas engine regulation. 

XII. Theory of Vapors .199 

Formation at constant pressure : saturated steam : superheated 
steam : paths of vapors : vapors in general : steam cycles : steam 
tables. 

vii 



Vlll 



CONTENTS 



XIII. The Steam Engine 256 

Practical modifications of the Rankine cycle: complete and in- 
complete expansion, wiredrawing, cylinder condensation, ratio 
of expansion, the steam jacket, use of superheated steam, actual 
expansion curve, mean effective pressure, back pressure, clear- 
ance, compression, valve action : the entropy diagram : cylinder 
feed and cushion steam, Boulvin's method, Reeve's method : 
multiple expansion : desirability of complete expansion, conden- 
sation losses in compound cylinders, Woolf engine, receiver 
engine, tandem and cross compounds, combined diagrams, de- 
sign of compound engines, governing, the drop controversy, 
intermediate compounds, binary vapor engine: engine tests: 
indicators, calorimeters, heat supplied, heat rejected, heat trans- 
fers : types of steam engine. 

XIV. The Steam Turbine . 317 

Conversion of heat into velocity : the turbine cycle, effects of fric- 
tion, rate of flow : efficiency in directing velocities : velocity com- 
pounding, pressure compounding : efficiency of the turbine : de- 
sign of impulse and pressure turbines : commercial types and 
applications. 

XV. Results of Trials of Steam Engines and Steam Turbines 350 
Economy, condensing and non-condensing, of various commercial 
forms with saturated and superheated steam : mechanical effi- 



XVI. The Steam Power Plant . <• . 

Fuels, combustion economy, air supply, boilers, theory of draft, 
fans, chimneys, stokers, heaters, superheaters, economizers, con- 
densers, pumps, injectors. 

XVII. Distillation . 

The still, evaporation in vacuo, multiple-effect evaporation. 

Fusion : 
Change of volume during change of state, pressure-temperature 
relation, latent heat of fusion of ice. 

Liquefaction of Gases : 
Pressure and cooling, critical temperature, cascade system, regen- 
erative apparatus. 

XVIII. Mechanical Refrigeration 

Air machines: reversed cycle, Bell-Coleman machine, dense air 
apparatus, coefficient of performance, Kelvin warming ma- 
chine: vapor-compression machines: the cycle, choice of fluid, 
tonnage rating, ice-melting effect, design of compressor : the 
absorption system : methods and fields of application : ice-mak- 
ing : commercial efficiencies. 



360 



380 



394 



CHAPTER I 

THE NATURE AND EFFECTS OF HEAT 

1. Heat as Motive Power. All artificial motive powers derive their 
origin from heat. Muscular effort, the forces of the waterfall, the wind, 
tides and waves, and the energy developed by the combustion of fuel, may 
all be traced back to reactions induced by heat. Our solid, liquid, and 
gaseous fuels are stored-up solar heat in the forms of hydrogen and carbon. 

2. Nature of Heat. We speak of bodies as "hot" or "cold," referring 
to certain impressions which they produce upon our senses. Common 
experimental knowledge regarding heat is limited to sensations of temper- 
ature. Is heat matter, force, motion, or position ? The old " caloric " 
theory was that "heat was that substance whose entrance into our bodies 
causes the sensation of warmth, and whose egress the sensation of cold." 
But heat is not a " substance " similar to those with which we are familiar, 
for a hot body weighs no more than one which is cold. The calorists 
avoided this difficulty by assuming the existence of a weightless material 
fluid, caloric. This substance, present in the interstices of bodies, it was 
contended, produced the effects of heat; it had the property of passing 
between bodies over any intervening distance. Friction, for example, de- 
creased the capacity for caloric; and consequently some of the latter 
"flowed out," as to the hand of the observer, producing the sensation of 
heat. Davy, however, in 1799, proved that friction does not diminish the 
capacity of bodies for containing heat, by rubbing together two pieces of 
ice until they melted. According to the caloric theory, the resulting water 
should have had less capacity for heat than the original ice : but the fact is 
that water has actually about twice the capacity for heat that ice has ; or, 
in other words, the specific heat of water is about 1.0, while that of ice is 
0.504. The caloric theory was further assailed by Eumford, who showed 
that the supply of heat from a body put under appropriate conditions was 
so nearly inexhaustible that the source thereof could not be conceived as 
being even an " imponderable " substance. The notion of the calorists 
was that the different specific heats of bodies were due to a varying capac- 
ity for caloric ; that caloric might be squeezed out of a body like water 
from a sponge. Rumford measured the heat generated by the boring of 
cannon in the arsenal at Munich. In one experiment, a gun weighing 

1 



APPLIED THERMODYNAMICS 



113.13 lb. was heated 70° P., although the total weight of borings produced 
was only 837 grains troy. In a later experiment, Rumford succeeded in 
boiling water by the heat thus generated. He argued that "anything 
which any insulated body or system of bodies may continue to furnish without 
limitation cannot possibly be a material substance." The evolution of heat, 
it was contended, might continue as indefinitely as the generation of 
sound following the repeated striking of a bell (1). 

Joule, about 1845, showed conclusively that mechanical energy 

alone sufficed for the production of heat, and that the amount of heat 

. - generated was always proportionate to the 

j-LL energy expended. A view of his apparatus 

II is given in Fig. 1, v and h being the verti- 

I — ^|JV-j^[p ! ^ , cal and horizontal sections, respectively, of 

the container shown at c. Water being 
placed in e, a rotary motion of the contained 
brass paddle wheel was caused by the de- 
scent of two leaden weights suspended by 
cords. The rise in temperature of the 






Q> 



D 



Fig. 1. Arts. 2, 30. — Joule's Apparatus. 



water was noted, the expended work (by the falling weights) com- 
puted, and a proper correction made for radiation. Similar experi- 
ments were made with mercury instead of water. As a result of 
his experiments, Joule reached conclusions which served to finally 
overthrow the caloric theory. 

3. Mechanical Theory of Heat. Various ancient and modern 
philosophers had conceded that heat was a motion of the minute 
particles of the body, some of them suggesting that such motion 



THE NATURE AND EFFECTS OF HEAT 3 

was produced by an "igneous matter." Locke denned heat as u a 
very brisk agitation of the insensible parts of the object, which pro- 
duces in us that sensation from which we denominate the object 
hot ; so [that] what in our sensation is heat, in the object is nothing 
but motion." Young argued, "If heat be not a substance, it must 
be a quality; and this quality can only be a motion." This is the 
modern conception. Heat is energy : it can perform work, or pro- 
duce certain sensations ; it can be measured by its various effects. 
It is regarded as " energy stored in a substance by virtue of the state 
of its molecular motion" (2). 

Conceding that heat is energy, and remembering the expression for energy, 
\ ?nv 2 , it follows that if the mass of the particle does not change, its velocity (molec- 
ular velocity) must change ; or if heat is to include potential energy, then the 
molecular configuration must change. The molecular vibrations are invisible, and 
their precise nature unknown. Rankine's theory of molecular vortices assumes a 
law of vibration which has led to some useful results. 

Since heat is energy, its laws are those generally applicable to energy, 
as laid down by Newton : it must have a commensurable value ; it must 
be convertible into other forms of energy, and they to heat; and the 
equivalent of heat energy, expressed in mechanical energy units, must be 
constant and determinable by experiment. 

4. Subdivisions of the Subject. The evolutions and absorptions 
of heat accompanying atomic combinations and molecular decompo- 
sitions are the subjects of thermochemistry. The mutual relations of 
heat phenomena, with the consideration of the laws of heat trans- 
mission, are dealt with in general physics. The relations between 
heat and mechanical energy are included in the scope of applied engi- 
neering thermodynamics, which may be defined as the science of the 
mechanical theory of heat. While thermodynamics is thus apparently 
only a subdivision of that branch of physics which treats of heat, the 
relations which it considers are so important that it may be regarded 
as one of the two fundamental divisions of pl^sics, which from this 
standpoint includes mechanics — dealing with the phenomena of 
ordinary masses — and thermodynamics — treating of the phenomena 
of molecules. 

5. Applications of Thermodynamics. The subject has far-reaching 
applications in physics and chemistry. In its mechanical aspects, it deals 



4 APPLIED THERMODYNAMICS 

with matters fundamental to the engineer. After developing the general 
laws and dwelling briefly upon ideal processes, we are to study the condi- 
tions affecting the efficiency and capacity of air, gas, and steam engines 
and the steam turbine; together with the economics of air compression, 
distillation, refrigeration, and gaseous liquefaction. The ultimate engi- 
neering application of thermodynamics is in the saving of heat, an appli- 
cation which becomes attractive when viewed in its just aspect as a saving 
of money and a mode of conservation of our material wealth. 

6. Temperature. A hot body, in common language, is one whose 
temperature is high, while a cold body is one low in temperature. Tem- 
perature, then, is a measure of the hotness of bodies. From a rise in tem- 
perature, we infer an accession of heat; or from a fall in temperature, 
a loss of heat. Temperature is not, however, a satisfactory measure of 
quantities of heat. A pound of water at 200° contains very much more 
heat than a pound of lead at the same temperature ; this may be demon- 
strated by successively cooling the bodies in a bath to the same final tem- 
perature, and noting the gain of heat by the bath. Furthermore, immense 
quantities of heat are absorbed by bodies in passing from the solid to the 
liquid or from the liquid to the vaporous conditions, without any change 
in temperature whatever. Temperature defines a condition of heat only. 
It is a measure of the capacity of the body for communicating heat to other 
bodies. Heat always passes from a body of relatively high temperature ; 
it never passes of itself from a cold body to a hot one. Wherever two 
bodies of different temperatures are in thermal juxtaposition, an inter- 
change of heat takes place ; the cooler body absorbs heat from the hotter 
body, no matter which contains initially the greater quantity of heat, 
until the two are at the same temperature, or in thermal equilibrium. 
Two bodies are at the same temperature when there is no tendency toward a 
transfer of heat between them. Measurements of temperature are in gen- 
eral based upon arbitrary scales, standardized by comparison with some 
physically established " fixed " point. One of these fixed temperatures is 
that minimum at which pure water boils when under normal atmospheric 
pressure of 14.697 lb. per square inch; viz. 212° F. Another is the 
maximum temperature of melting ice at atmospheric pressure, which is 
32° F. Our arbitrary scales of temperature cannot be expressed in terms 
of the fundamental physical units of length and weight. 

7- Measurement of Temperature. Temperatures are measured by thermome- 
ters. The common type of instrument consists of a connected bulb and vertical 
tube, of glass, in which are contained a liquid. Any change in temperature affects 
the volume of the liquid, and the portion in the tube consequently rises or falls. 
The expansion of solids or of gases is sometimes utilized in the design of thermom- 



THE NATURE AND EFFECTS OF HEAT 5 

eters. Mercury and alcohol are the liquids commonly used. The former freezes at 
- 38° F. and boils at 675° F. The latter freezes at - 203° F. and boils at 173° F. 
The mercury thermometer is, therefore, more commonly used for high tempera- 
tures, and the alcohol for low. 

8. Thermometric Scales. The Fahrenheit thermometer, generally 
employed by engineers in the United States and Great Britain, 
divides the space between the "fixed points" (Art. 6) into 180 
equal degrees, freezing being at 32° and boiling at 212°. The 
Centigrade scale, employed by chemists and physicists (sometimes 
described as the Celsius scale), calls the freezing point 0° and the 
boiling point 100°. On the Reaumur scale, used in Russia and a 
few other countries, water freezes at 0° and boils at 80°. One de- 
gree on the Fahrenheit scale is, therefore, equal to -|° C, or to |° R. 
In making transformations, care must be taken to regard the differ- 
ent zero point of the Fahrenheit thermometer. On all scales, tem- 
peratures below zero are distinguished by the minus ( — ) prefix. 

The Centigrade scale is unquestionably superior in facilitating arithmetical 
calculations; but as most English papers and tables are published in Fahrenheit 
units, we must, for the present at least, use that scale of temperatures. 

9. High Temperature Measurements. For measuring temperatures above 
800° F., some form of pyrometer must be employed. The simplest of these is the 
metallic pyrometer, exemplifying the principle that different metals expand to dif- 
ferent extents when heated through the same range of temperature. Bars of iron 
and brass are firmly connected at one end, the other ends being free. At some 
standard temperature the two bars are of the same length, and the indicator, con- 
trolled jointly by the two free ends of the bars, registers that temperature. When 
the temperature changes, the indicator is moved to a new position by the relative 
distortion of the free ends. 

In the Le Chatelier electric pyrometer, a thermoelectric couple is employed. For 
temperatures ranging from 300° C. to 1500° C, one element is made of platinum, 
the other of a 10 per cent, alloy of platinum with rhodium. Any rise in tempera- 
ture at the junction of the elements induces a flow of electric current, which is con- 
ducted by wires to a galvanometer, located in any convenient position. The ex- 
pensive metallic elements are protected from oxidation by enclosing porcelain 
tubes. In the Bristol thermoelectric instrument, one element is of a platinum- 
rhodium alloy, the other of a cheaper metal. The electromotive force is indicated 
by a Weston millivoltmeter, graduated to read temperatures directly. The in- 
strument is accurate up to 2000° F. The electrical resistance pyrometer is based on 
the law of increase of electrical resistance with increase of temperature. In Cal- 
endar's form, a coil of fine platinum wire is wound on a serrated mica frame. 
The instrument is enclosed in porcelain, and placed in the space the temperature 



6 APPLIED THERMODYNAMICS 

of which is to be ascertained. The resistance is measured by a Wheatstone bridge, 
a galvanometer, or a potentiometer, calibrated to read temperatures directly. 
Each instrument must be separately calibrated. 

Optical pyrometers are based on the principle that the colors of bodies vary 
with their temperatures. In the Morse thermogage, of this type, an incandescent 
lamp is wired in circuit with a rheostat and a millivoltmeter. The lamp is located 
between the eye and the object, and the current is regulated until the lamp be- 
comes invisible. The temperature is then read directly from the calibrated milli- 
voltmeter. The device is extensively used in hardening steel tools, and has been 
employed to measure the temperatures in steam boiler furnaces. 

10. Cardinal Properties. A cardinal or integral property of a 
substance is any property which is fully defined by the immediate 
state of the substance. Thus, weight, length, specific gravity, are 
cardinal properties. On the other hand, cost is a non-cardinal prop- 
erty ; the cost of a substance cannot be determined by examination 
of that substance; it depends upon the previous history of the sub- 
stance. Any two or three cardinal properties of a substance may be 
used as coordinates in a graphic representation of the state of the sub- 
stance. Properties not cardinal may not be so used, because such 
properties do not determine, nor are they determinable by, the pres- 
ent state of the substance. The cardinal properties employed in 
thermodynamics are five or six in number.* Three of these are pres- 
sure, volume, and temperature ; pressure being understood to mean 
specific pressure, or uniform pressure per unit of surface, exerted by or 
upon the body, and volume to mean volume per unit of weight. The 
location of any point in space is fully determined by its three coordi- 
nates. Similarly, any three cardinal properties may serve to fix the 
thermal condition of a substance. 

The first general principle of thermodynamics is that if two of the 
three named cardinal properties are known, these two enable us to calcu- 
late the third. This principle cannot be proved a priori ; it is to be justi- 
fied by its results in practice. Other thermodynamic properties than 
pressure, volume, and temperature conform to the same general principle 
(Art. 169) ; with these properties we are as yet unacquainted. A correlated 
principle is, then, that any two of the cardinal properties suffice to fully 
determine the state of the substance. For certain gases, the general prin- 
ciple may be expressed, PV= ( f)T 

* For gases, pressure, volume, temperature, internal energy, entropy ; for wet 
vapors, these five and dryness. 



THE NATURE AND EFFECTS OF HEAT 7 

while for other gaseous fluids more complex equations (Art. 363) must be 
used. In general, these equations are, in the language of analytical geom- 
etry, equations to a surface. Certain vapors cannot be represented, as 
yet, by any single equation between P, V> and T, although corresponding 
values of these properties may have been ascertained by experiment. 

11. Preliminary Assumptions. The greater part of the subject 
deals with substances assumed to be in a state of mechanical equilibrium, 
all changes being made with infinite slowness. A second assumption 
is that no chemical actions occur during the thermodynamic trans- 
formation. In the third place, the substances dealt with are assumed 
to be so homogeneous, as to be in uniform thermal condition through- 
out : for example, the pressure property must involve equality of 
pressure in all directions ; and this limits the consideration to the 
properties of liquids and gases. 

The thermodynamics of solids is extremely complex, because of the obscure 
stresses accompanying their deformation (3). Kelvin (4) has presented a general 
analysis of the action of any homogeneous solid body homogeneously strained. 

12. The Three Effects of Heat. Setting aside the obvious un- 
classified changes in pressure, volume, and temperature accompanying 
manifestations of heat energy, there are three known ways in which 
heat may be expended. They are : 

(#) In a change of temperature of the substance. 

(5) In a change of physical state of the substance. 

(<?) In the performance of external work by or upon the substance. 
Denoting these effects by T, J, and W, then, for any transfer of heat 
H, we have the relation 

H=T + I + W, 

any of the terms of which expression may be negative. It should be 
quite obvious, therefore, that changes of temperature alone are in- 
sufficient to measure expenditures of heat. 

Items (a) and (b) are sometimes grouped together as indications 
of a change in the INTERNAL ENERGY of the heated substance, the 
term being one of the first importance, which it is essential to clearly 
apprehend. Items (5) and (<?) are similarly sometimes combined as 
representing the total work. 



8 APPLIED THERMODYNAMICS 

13. The Temperature Effect. Temperature indications of heat activity are 
sometimes referred to as " sensible heat." The addition of heat to a substance 
may either raise or lower its temperature, in accordance with the fundamental 
equation of Art. 12. 

The temperature effect of heat, from the standpoint of the mechanical 
theory, is due to a change in the velocity of molecular motion, in conse- 
quence of which the kinetic energy of that motion changes. 

This effect is therefore sometimes referred to as vibration work. Clausius 
called it actual energy. 

14. External Work Effect. The expansion of solids and fluids, due to the supply 
of heat, is a familiar phenomenon. Heat may cause either expansion or contraction, 
which, if exerted against a resistance, may suffice to perform mechanical work. 

15. Changes of Physical State. Broadly speaking, such effects 
include all changes, other than those of temperature, within the sub- 
stance itself. The most familiar examples are the change between 
the solid and the liquid condition, when the substance melts or 
freezes, and that between the liquid and the vaporous, when it boils 
or condenses; but there are intermediate changes of molecular aggrega- 
tion in all material bodies which are to be classed with these effects 
under the general description, disgregation work. The mechanical 
theory assumes that in such changes the molecules are moved into 
new positions, with or against the lines of mutual attraction. These 
movements are analogous to the "partial raising or lowering of a 
weight which is later to be caused to perform work by its own descent. 
The potential energy of the substance is thus changed, and positive 
or negative work is performed against internal resisting forces." 

When a substance changes its physical state, as from water to steam, it 
can be shown that a very considerable amount of external work is done, in 
consequence of the increase in volume which occurs, and which may be 
made to occur against a heavy pressure. This external work is, however, 
equivalent only to a very small proportion of the total heat supplied to 
produce evaporation, the balance of the heat having been expended in the 
performance of disgregation work. 

The molecular displacements constituting disgregation work are exemplified in 
the phenomena of solution, and in the action of freezing mixtures (5). 

16. Solid, Liquid, Vapor, Gas. Solid bodies are those which resist tendencies 
to change their form or volume. Liquids are those bodies which in all of their 



THE NATURE AND EFFECTS OF HEAT 9 

parts tend to preserve definite volume, and which are practically unresistant to 
influences tending to slowly change their figure. Gases are unresistant to slow 
changes in figure or to increases in volume. They tend to expand indefinitely so 
as to completely fill any space in which they are contained, no matter what the 
shape or the size of that space may be. Most substances have been observed in 
all three forms, under appropriate conditions ; and all substances can exist in any 
of the forms. At this stage of the discussion, no essential difference need be 
drawn between a vapor and a gas. Formerly, the name vapor was applied to 
those gaseous substances which at ordinary temperatures were liquid, while a 
" gas " was a substance never observed in the liquid condition. Since all of the 
so-called "permanent" gases have been liquefied, this distinction has lost its force. 
A useful definition of a vapor as distinct from a true gas will be given later 
(Art. 380). 

Under normal atmospheric pressure, there exist well-defined tempera- 
tures at which various substances pass from the solid to the liquid and 
from the liquid to the gaseous conditions. The temperature at which the 
former change occurs is called the melting point or freezing point; that of 
the latter is known as the boiling point or temperature of condensation. 

17. Other Changes of State. Although the operation described as boiling 
occurs, for each liquid, at some definite temperature, there is an almost continual 
evolution of vapor from nearly all liquids at temperatures below their boiling 
points. The water of the earth's surface, for example, is slowly changing to vapor 
and impregnating the atmosphere. Such " insensible " evaporation is with some 
substances non-existent, or at least too small in amount to permit of measure- 
ment : as in the instances of mercury at 32° F. or of sulphuric acid at any ordi- 
nary temperature. Ordinarily, a liquid at a given temperature continues to 
evaporate so long as its partial vapor pressure is less than the maximum pressure 
corresponding to its temperature. The interesting phenomenon of sublimation 
consists in the direct passage from the solid to the gaseous state. Such sub- 
stances as camphor and iodine manifest this property. Ice and snow also pass 
directly to a state of vapor at temperatures far below the freezing point. There 
seem to be no quantitative data on the heat relations accompanying this change 
of state. 

18. Variations in "Fixed Points." Aside from the influence of pressure 
(Arts. 358, 603), various causes may modify the positions of the "fixed points " of 
the thermometric scale. Water may be cooled below 32° F, without freezing, 
if kept perfectly still. If free from air, water boils at 270-290° F. Minute par- 
ticles of air are necessary to start evaporation sooner ; their function is probably 
to aid in the diffusion of heat. 

(1) Tyndall : Heat as a Mode of Motion. (2) Nichols and Franklin : The Ele- 
ments of Physics, I, 161. (3) See paper by J. E. Siebel : The Molecular Constitu- 
tion of Solids, in Science, Nov. 6, 1909, p. 654. (4) Quarterly Mathematical Journal, 
April, 1855. (5) Darling: Heat for Engineers, 208. 



10 APPLIED THERMODYNAMICS 



SYNOPSIS OF CHAPTER I 

Heat is the universal source of motive power. 

Theories of heat : the caloric theory — heat is matter ; the mechanical theory — heat 

is molecular motion, mutually convertible with mechanical energy. 
Thermochemistry, Thermodynamics. 
Thermodynamics : the mechanical theory of heat ; in its engineering applications, the 

science of heat-motor efficiency. 
Heat intensity, temperature : definition of, measurement of ; pyrometers. 
Thermometric scales : Fahrenheit, Centigrade, Reaumur ; fixed points and their 

variations. 
Cardinal properties : pressure, volume, temperature ; TV— (/) T. 
Assumptio7is : uniform thermal condition ; no chemical action ; mechanical equilibrium . 
Effects of heat : H = T-\- I + W; T 4- 1 ' = "internal energy" ; W = external work. 
Changes of physical state, perceptible and imperceptible : I— disgregation work. 
Solid, liquid, vapor, gas : melting point, boiling point ; insensible evaporation ; 

sublimation. 

PROBLEMS 

1. Compute the freezing points, on the Centigrade scale, of mercury and alcohol. 

2. At what temperatures, Reaumur, do alcohol and mercury boil ? 

3. The normal temperature of the human body is 98.6° F. Express in Centigrade 
degrees. 

4. At what temperatures do the Fahrenheit and Centigrade thermometers read 
alike ? 

5. At what temperatures do the Fahrenheit and Reaumur thermometers read 
alike ? 

6. Express the temperature — 273° C. on the Fahrenheit and RCaumur scales. 



CHAPTER II 

THE HEAT UNIT: SPECIFIC HEAT: FIRST LAW OF 
THERMODYNAMICS 

19. Temperature — Waterfall Analogy. The difference between temperature 
and quantity of heat may be apprehended from the analogy of a waterfall. Tem- 
perature is like the head of water ; the energy of the fall depends upon the head, 
but cannot be computed without knowing at the same time the quantity of water. 
As waterfalls of equal height may differ in power, while those of equal power may 
differ in fall, so bodies at like temperatures may contain different quantities of 
heat, and those at unequal temperatures may be equal in heat contents. 

20. Temperatures and Heat Quantities. If we mix equal weights of 
water at different temperatures, the resulting temperature of the mix- 
ture will be very nearly a mean between the two initial temperatures. 
If the original weights are unequal, then the final temperature will be 
nearer that initially held by the greater weight. The general principle of 
transfer is that 

The loss of heat by the hotter water will equal the gain of heat by the 
colder. 

Thus, 5 lb. of water at 200° mixed with 1 lb. at 104° gives 6 lb. at 
184°; the hotter water having lost 80 "pound-degrees," and the colder 
water having gained the same amount of heat. If, however, we mix the 
5 lb. of hot water with. 1 lb. of some other substance — say linseed oil — 
the resulting temperature will not be 184°, but 194.6°, if the initial tem- 
perature of the oil is 104°. 

21. General Principles. Before proceeding, we may note, in addition to the 
principle just laid down, the following laws which are made apparent by the ex- 
periments described and others of a similar nature : 

(a) In a homogeneous substance, the movement of heat accom- 
panying a given change of temperature * is proportional to the 
weight of the substance. 

(6) The movement of heat corresponding to a given change of 

* Not only the amount, hut the method, of changing the temperature must be 
fixed (Art. 57). 

11 



12 APPLIED THERMODYNAMICS 

temperature is not necessarily the same for equal intervals at all 
parts of the thermometric scale ; thus, water cooling from 200° to 
195° does not give out exactly the same quantity of heat as in cool- 
ing from 100° to 95°. 

(c) The loss of heat during cooling through a stated range of 
temperature is exactly equal to the gain of heat during warming 
through the same range. 

22. The Heat Unit. Changes of temperature alone do not measure heat quan- 
tities, because heat produces other effects than that of temperature change. If, 
however, we place a body under "standard" conditions, at which these other 
effects, if not known, are at least constant, then we may define a unit of quantity 
of heat by reference to the change in temperature which it produces, understand- 
ing that there may be included perceptible or imperceptible changes of other 
kinds, not affecting the constancy of value of the unit. 

The British Thermal Unit is that quantity of heat which is expended in 
raising the temperature of one pound of water (or in producing other effects 
during this change in temperature) from 62° to 63° F.* 

To heat water over this range of temperature requires very nearly the same 
expenditure of heat as is necessary to warm it 1° at any point on the thermometric 
scale. In fact, some writers define the heat unit as that quantity of heat necessary 
to change the temperature from 39.1° (the temperature of maximum density) to 
40.1°. Others use the ranges 32° to 33°, 59° to 60°, or 39° to 40°. The range" first 
given is that most recently adopted. 

23. French Units. The French or C. G. S. unit of heat is the 
calorie, the amount of heat necessary to raise the temperature of one 
kilogram of water 1° C. Its value is 2.2046 x f = 3.96832 B. t. u., and 
1 B. t. u. = 0.251996 cal. The calorie is variously measured from 4° to 
5° and from 14.5° to 15.5° C. The gram-calorie is the heat required to 
raise the temperature of one gram of water 1° C. The Centigrade heat 
unit measures the heat necessary to raise one pound of water 1° C. in 
temperature. 

24. Specific Heat. Reference was made in Art. 20 to the different heat 
capacities of different substances, e.g. water and linseed oil. If Ave mix 
a stated quantity of water at a fixed temperature successively with equal 
weights of various materials, all initially at the same temperature, the 
final temperatures of the mixtures will all differ, indicating that a unit 

* There are certain grounds for preferring that definition which makes the B. t. u. 
t'.ie j\q part of the amount of heat required to raise the temperature of one pound of 
water at atmospheric pressure from the freezing point to the boiling point. 



THE HEAT UNIT. SPECIFIC HEAT 13 

rise of temperature of unit weight of these various materials represents a 
different expenditure of heat in each case. 

The property by virtue of which materials differ in this respect is 
that of specific heat, which may be denned as the quantity of heat 
necessary to raise the temperature of unit weight of a body through one 
degree. 

The specific heat of water at standard temperature (Art. 22) is, meas- 
ured in B. t. u., 1.0 ; generally speaking, its value is slightly variable, as is 
that of all substances. 

Rankine's definition of specific heat is illustrative : " the specific heat of any 
substance is the ratio of the weight of water at or near 39.1° F. [62°-63° F.] which 
has its temperature altered one degree by the transfer of a given quantity of heat, 
to the weight of the other substance under consideration, which has its temperature 
altered one degree by the transfer of an equal quantity of heat." 

25. Mixtures of Different Bodies. If the weights of a group of 
mixed bodies be X, Y, Z, etc., their specific heats x, y, z, etc., their ini- 
tial temperatures t, u, v, etc., and the final temperature of the mixture 
be m, then we have the following as a general equation of thermal equi- 
librium, in which any quantity may be solved for as an unknown: 

xX(t — m)-\-y Y(u — m) + zZ(v — m) • • • =0. 

This illustrates the usual method of ascertaining the specific heat of any 
body. When all the specific heats are known, the loss of heat to sur- 
rounding bodies may be ascertained by introducing the additional term, 
-f- R, on the left-hand side of this equation. The solution will usually 
give a negative value for R, indicating that surrounding bodies have 
absorbed rather than contributed heat. The value of R will of course be 
expressed in heat units. 

26. Specific Heat of Water. The specific heat of water, according 
to Rowland's experiments, decreases as the temperature is increased 
from 39.1° to 80° F., at which latter temperature it reaches a minimum 
value, afterward increasing (Art. 359, footnote). The variation in its 
value is very small. The approximate specific heat, 1.0, is high as com- 
pared with that of almost all other substances. 

27. Problems Involving Specific Heat. The quantity of heat re- 
quired to produce a given change of temperature in a body is equal 
to the weight of the body, multiplied by the range of temperature 
and by the specific heat. 

Or, symbolically, using the notation of Art. 25, 

H= xX(m -t). 



14 APPLIED THERMODYNAMICS 

If the body is cooled, then m, the final temperature, is less than t, and the sign of 
H is - ; if the body is warmed, the sign of II is + , indicating a reception of heat. 

28. Consequences of the Mechanical Theory. The Mechanical Equivalent 
of Heat. Even before Joule's formulation (Art. 2), Rumford's ex- 
periments had sufficed for a comparison of certain effects of heat 
with an expenditure of mechanical energy. The power exerted by the 
Bavarian horses used to drive his machinery is uncertain ; but Alexander 
has computed the approximate relation to have been 847 foot-pounds = 
1 B. t. u. (1), while another writer fixes the ratio at 1034, and Joule cal- 
culated the value obtained to have been 849. 

Carnot's work, although based throughout on the caloric theory, shows evident 
doubts as to its validity. This writer suggested (1824) a repetition of Rumford's 
experiments, with provision for accurately measuring the force employed. Using 
a method later employed by Mayer (Art. 29) he calculated that "0.611 units 
of motive power" were equivalent to "550 units of heat"; a relation which 
Tyndall computes as representing 370 kilogram-meters per calorie, or 676 foot- 
pounds per B. t. u. Montgolfier and Seguin (1839) may possibly have anticipated 
Mayer's analysis. 

29. Mayer's Calculation. This obscure German physician published in 1842 
(2) his calculation of the mechanical equivalent of heat, based on the difference 
in the specific heats of air at constant pressure and constant volume, giving 
the ratio 771.4 foot-pounds per B. t. u. (Art. 72). This was a substantially correct 
result, though given little consideration at the time. Mayer had previously made 
rough calculations of equivalence, one being based on the rise of temperature 
occurring in the " beaters " of a paper mill. 

30. Joule's Determination. Joule, in 1843, presented the first of his 
exhaustive papers on the subject. The usual form of apparatus employed 
has been shown in Fig. 1. In the appendix to his paper Joule gave 770 as 
the best value deducible from his experiments. In 1849 (3) he presented 
the figure for many years afterward accepted as final, viz. 772. 

In 1878 an entirely new set of experiments led to the value 772.55, which 
Joule regarded as probably slightly too low. Experiments in 1857 had given the 
values 745, 753, and 766. Most of the tests were made with water at about 60° F. 
This, with the value of g at Manchester, where the experiments were made, in- 
volves slight corrections to reduce the results to standard conditions (4). 

31. Other Investigators. Of independent, though uncertain, merit, were the 
results deduced by the Danish engineer, Colding, in 1843. His value of the 
equivalent is given by Tyndall as 638 (5). Helmholtz (1847) treated the matter 
of equivalence from a speculative standpoint. Assuming that " perpetual motion " 
is impossible, he contended that there must be a definite relation between heat 
energy and mechanical energy. As early as 1845, Holtzmann (6) had apparently 



MECHANICAL EQUIVALENT OF HEAT 15 

independently calculated the equivalence by Mayer's method. By 1847 the reality 
of the numerical relation had been so thoroughly established that little more was 
heard of the caloric theory. Clausius, following Mayer, in 1850 obtained wide 
circulation for the value 758 (7) . 

32. Hirn's Investigation. Joule had employed mechanical agencies in the 
heating of water. Hirn, in 1865 (8), described an experiment by which he trans- 
formed into heat the work expended in producing the impact of solid bodies. 
Two blocks, one of iron, the other of wood, faced with iron in contact with a lead 
cylinder, were suspended side by side as pendulums. The iron block was allowed 
to strike against the wood block and the rise in temperature of w r ater contained in 
the lead cylinder was noted and compared with the computed energy of impact. 
The value obtained for the equivalent was 775. 

Far more conclusive, though less accurate, results were obtained 
by Hirn by noting that the heat in the exhaust steam from an engine 
cylinder was less than that which was present in the entering steam. 
It was shoAvn by Clausius that the heat which had disappeared was 
always roughly proportional to the work done by the engine, the 
average ratio of foot-pounds to heat units being 753 to 1. This was 
virtually a reversal of Joule's experiment, illustrating as it did the 
conversion of heat into work. It is the most striking proof we have 
of the equivalence of work and heat. 

33. Recent Practice. In 1876 a committee of the British Association for the 
Advancement of Science reviewed critically the work of Joule, and as a mean 
value, derived from his best 60 experiments, recommended the use of the figure 
774.1, which was computed to be correct within 4^. In 1879, Rowland, having 
conducted exact experiments on the specific heat of water, carefully redetermined 
the value of the equivalent by driving a paddle wheel about a vertical axis at 
fixed speed, in a vessel of water prevented from turning by counterbalance weights. 
The torque exerted by the paddle was measured. This permitted of a calculation 
of the energy expended, which was compared with the rise in temperature of the 
water. Rowland's value was 778, with water at its maximum density. This 
was regarded as possibly slightly low -(9). Since the date of Rowland's work, the 
subject has been investigated by Griffiths (10), who makes the value somewhat 
greater than 778, and by Reynolds and Moorby (11), who report the ratio 778 as 
the mean obtained for a range of temperature from 32° to 212° F. This they 
regard as possibly 1 or 2 foot-pounds too low. 

34. Summary. The establishing of a definite mechanical equivalent of 
heat may be regarded as the foundation stone of thermodynamics. Accord- 
ing to Merz (12), the anticipation of such an equivalent is due to Poncelet 
and Carnot ; Rumford's name might be added. " The first philosophical 
generalizations were given by Mohr and Mayer; the first mathematical 



16 APPLIED THERMODYNAMICS 

treatment by Helmholtz ; the first satisfactory experimental verification 
by Joule." The construction of the modern science on this foundation 
has been the work chiefly of Rankine, Clausius, and Kelvin. 

35. First Law of Thermodynamics. Heat and mechanical energy- 
are mutually convertible in the ratio of 778 foot-pounds to the British 
thermal unit. 

This is a restricted statement of the general principle of the conservation of 
energy, a principle which is itself probably not susceptible to proof. 
We have four distinct proofs of the first law : 

(a) Joule's and Rowland's experiments on the production of 
heat by mechanical work. 

(6) Hirn's observations on the production of work by the ex- 
penditure of heat. 

(V) The computations of Mayer and others, from general data. 

(cT) The fact that the law enables us to predict thermal proper- 
ties of substances which experiments confirm. 

36. Wormell's Theorem. There cannot be two values of the mechanical 
equivalent of heat. Consider two machines, A and B, in the first of which work 
is transformed into heat, and in the second of which heat is transformed into 
work. Let / be the mechanical equivalent of heat for A, W the amount of work 
which it consumes in producing the heat Q; then W = JQ or Q = W -4- J. Let 
this heat Q be used to drive the machine B, in which the mechanical equivalent 
of heat is, say K. Then the work done by B is V = KQ = KW -s- J. Let this 
work be now expended in driving A. It will produce heat R, such that JR = V 
or R = V -f- J. If this heat R be used in B, work will be done equal to KR ; but 

KR = KV + J = (K\* W. 

Similarly, after n complete periods of operation, all parts of the machines occupy- 
ing the same positions as at the beginning, the work ultimately done by B will be 



(!)' 



w. 



If K is less than /, this expression will decrease as n increases ; i.e. the system 
will tend continually to a state of rest, contrary to the first law of motion. If K 
be greater than J, then as n increases the work constantly increases, involving the 
assumed fallacy of perpetual motion. Hence K and J must be equal (13). 

37. Significance of the Mechanical Equivalent. A very little heat is seen to be 
equivalent to a great deal of work. The heat used in raising the temperature of 
one pound of water 100° represents energy sufficient to lift one ton of water nearly 
39 feet. The heat employed to boil one pound of water initially at 32° F. would 



FIRST LAW OF THERMODYNAMICS 17 

suffice to lift one ton 443 feet. The heat evolved in the combustion of one pound of 
hydrogen (62,000 B. t. u.) would lift one ton nearly five miles. 

(1) Treatise on Thermodynamics, London, 1892. (2) Wohler and Liebig's 
Annalen der Pharmacie : Bemerkungen uber die Krdfte der unbelebten Natur, May, 
1842. (3) Phil. Trans., 1850. (4) Joule's Scie?itific Papers, Physical Society of 
London, 1884. (5) Probably quoted by Tyndall from a later article by Colding, in 
which this figure is given. Colding's original paper does not seem to be accessible. 

(6) Ueber die Warme und Elasticitdt der G-ase und Dampfe, Mannheim, 1845. 

(7) Poggendorff, Annalen, 1850. (8) Theorie Mecamque, etc., Paris, 1865. (9) Proc. 
Amer. Acad. Arts and Sciences, New Series, VII, 1878-79. (10) Phil. Trans. Boy. 
Soc, 1893. (11) Phil. Trans., 1897. (12) History of European Thought, II, 187. 
(13) R. Wormell : Thermodynamics, 1886. 



SYNOPSIS OF CHAPTER II 

Heat and temperature : heat quantity vs. heat intensity. 

Principles : (a) heat movement proportional to weight of substance ; (6) temperature 
range does not accurately measure heat movement ; (c) loss during cooling equals 
gain during warming, for identical ranges. 

The British thermal unit : other units of heat quantity. 

Specific heat : mixtures of bodies ; quantity of heat to produce a given change of tern- 
perature ; specific heat of water. 

The mechanical equivalent of heat : early approximations. First law of thermody- 
namics : proofs ; only one value possible ; examples of the motive power of heat. 



PROBLEMS 

1. How many Centigrade heat units are equivalent to one calorie ? 

2. Find the number of gram-calories in one B.t. u. 

3. A mixture is made of 5 lb. of water at 200°, 3 lb. of linseed oil at 110°, and 
22 lb. of iron at 220°, the respective specific heats being 1.0, 0.3, and 0.12. Find the 
final temperature, if no loss occurs by radiation. 

4. If the final temperature of the mixture in Problem 3 is 189° F., find the num- 
ber of heat units lost by radiation. 

5. Under what conditions, in Problem 3, might the final temperature exceed that 
computed ? 

6. How much heat is given out by 7| lb. of linseed oil in cooling from 400° F. to 
32° F. ? 

7. In a heat engine test, each pound of steam leaves the engine containing 125.2 
B. t. u. less heat than when it entered the cylinder. The engine develops 155 horse- 
power, and consumes 3160 lb. of steam per hour. Compute the mechanical equivalent 
of heat. 

8. A pound of good coal will evolve 14,000 B.t. u. Assuming a train resistance 
of- 11 lb. per ton of train load, how far should one ton (2000 lb.) of coal, burned in the 
locomotive without loss, propel a train weighing 2000 tons ? If the locomotive weighs 
125 tons, how high would one pound of coal lift it, if fully utilized ? 



18 APPLIED THERMODYNAMICS 

9. Find the number of kilogram-meters equivalent to 1 calorie. (1 meter = 39.37 
in., 1 kilogram = 2.2046 lb.) 

10. Transform the following formula (P being the pressure in kilograms per square 
meter, Fthe volume in cubic meters per kilogram, Tthe Centigrade temperature plus 
273), to English units, letting the pressure be in pounds per square inch, the volume 
in cubic feet per pound, and the temperature that on the Fahrenheit scale plus 459.4, 
and eliminating coefficients in places where they do not appear in the original equation : 

PV= 47.1 T- P(l + 0.000002 P) fo.03l(— 3 ^ - 0.0052"! . 



CHAPTER III 

LAAYS OF GASES : ABSOLUTE TEMPERATURE : THE PERFECT GAS 

38. Boyle's (or Mariotte's) Law. The simplest thermodynamic 
relations are those exemplified by the so-called permanent gases. 
Boyle (Oxford, 1662) and Mariotte (1676-1679) separately enun- 
ciated the principle that at constant temperature the volumes of gases 
are inversely proportional to their pressures. In other words, the 
product of the specific volume and the pressure of a gas at a given 
temperature is a constant. For air, which at 32° F. has a volume 
of 12.387 cubic feet per pound when at normal atmospheric pressure, 
the value of the constant is, for this temperature^ 

141x14.7 x 12.387 = 26,221. 

Symbolically, if c denotes the constant for any given tempera- 
ture, 

pv = P V or, pv = c. 

Figure 2 represents Boyle's law graphically, the ordinates being pres- 
sures per square foot, and the abscissas, volumes in cubic feet per pound. 
The curves are a series of equilateral hyperbolas, plotted from the second 
of the equations just given, with various values of c. 

39. Deviations from Boyle's Law. This experimentally determined principle 
was at first thought to apply rigorously to all true gases. It is now known to be 
not strictly correct for any of them, although very nearly so for air, hydrogen, 
nitrogen, oxygen, and some others. All gases may be liquefied, and all liquids 
may be gasified. When far from the point of liquefaction, gases follow Boyle's 
law. When brought near the liquefying point by the combined influences of high 
pressure and low temperature, they depart widely from it. The four gases just 
mentioned ordinarily occur at far higher temperatures than those at which they 
will liquefy. Steam, carbon dioxide, ammonia vapor, and some other well-known 
gaseous substances which may easily be liquefied do not follow the law even 
approximately. Conformity with Boyle's law may be regarded as a measure of 
the " perfectness " of a gas, or of its approximation to the truly gaseous condition. 

19 



p 




































8U0U 








































e 


































C 




































\ 


' 


g 






























COCO 






\ 
































\ 




\ 


































\\ 


\ 






























A900 




\\ 


\ 
































IV 




































\\ 


\ 


































\\ 


X 




?\ 






























<Ts' 






aS. 




























t 








































^. 




























































rf— 


























d 











10 20 30 40 50 60 

Fig. 2. Arts. 38, 91. — Boyle's Law. 



40. Dalton's Law, Avogadro's Principle. Dalton has been credited (though 
erroneously) with the announcement of the law now known as that of Gay-Lussac 
or Charles (Art, 41). What is properly known as Dalton's law may be thus 
stated : A mixture of gases having no chemical action on one another exerts a pres- 
sure which is the sum of the pressures which would be exerted by the component 
gases separately if each in turn occupied the containing vessel alone at the given 
temperature. 

The ratio of volumes, at standard temperature and pressure, in which two 
gases combine chemically is always a simple rational fraction Q, f, f, etc.). 
Taken in conjunction with the molecular theory of chemical combination, this 
law leads to the principle of Avogadro that all gases contain the same number of 
molecules per unit of volume, at the same temperature and pressure. This law has 
important thermodynamic relations. 



41. Law of Gay-Lussac or of Charles (1). Davy had announced that the 
coefficient of expansion of air was independent of the pressure. Gay-Lus- 
sac verified this by the apparatus shown in Fig. 3. He employed a glass 
tube with a large reservoir A, containing the air, which had been previously 



LAWS OF GASES 



21 



dried. An index of mercury mn separated the air from the external atmos- 
phere, while permitting it to expand. The vessel B was first filled with 
melting ice. Upon applying heat, equal in- 
tervals of temperature shown on the ther- 
mometer G were found to correspond with 
equal displacements of the index mn. When 
a pressure was applied on the atmospheric 
side of the index, the proportionate expansion 
of the air was shown to be still constant for 
equal intervals of temperature, and to be equal 
to that observed under atmospheric pressure. 
Precisely the same results were obtained with Fig. 3. Arts. 41,48. —Verifica- 

,i m, • pi • tion of Charles' Law. 

other gases. The expansion of dry air was 

found to be 0.00375, or yi T of the volume at the freezing point, for each 

degree C. of rise of temperature. The law thus established may be 

expressed : 

For all gases, and at any pressure, maintained constant, equal increments of 
volume accompany equal increments of temperature. 





42. Increase of Pressure at Constant Volume. A second statement 
of this law is that all gases, when maintained at constant volume, 

undergo equal increases of 
pressure with equal increases 
of temperature. 

This is shown experimen- 
tally by the apparatus of Fig. 4. 
The glass bulb A contains the 
gas. It communicates with the 
open tube manometer Mm, 
which measures the pressure 
P is a tube containing mercury, 
in which an iron rod is submerged to a sufficient depth to keep the level 
of the mercury in m at the marked point a, thus maintaining a constant 
volume of gas. 

43. Regnault's Experiments. The constant 0.00375 obtained by G-ay- 
Lussac was pointed out by E-udberg to be probably slightly inaccurate. 
Kegnault, by employing four distinct methods, one of which was sub- 
stantially that just described, determined accurately the coefficient of 
increase of pressure, and finally the coefficient of expansion at constant 
pressure, which for dry air was found to be 0.003665, or ^l^, per degree 
C, of the volume at the freezing point. 



Fig. 4. Arts. 42, 48. — Coefficient of Pressure 



/ 

V 



22 APPLIED THERMODYNAMICS 

44. Graphical Representation. In Fig. 5, let ab represent the 
volume of a pound of gas at 32° F. Let temperatures and volumes 

be represented, respectively, by ordinates and 

/ e abscissas. According to Charles' Law, if the 

/ pressure be constant, the volumes and tempera- 

-J- v tures will increase proportionately ; the volume 

ab increasing ^3 for each degree C. that the 

temperature is increased, and vice versa. The 

straight line cbe then represents the successive 

relations of volume and temperature as the gas 

Fig. 5. Arts. 44, 84.— is heated or cooled from the temperature at b. 

Charles' Law. ^ the point e, where this line meets the a T axis, 

the volume of the gas will be zero, and its temperature will be 273° C, 

or 491.4° F., beloiv the freezing point. 

45. Absolute Zero. This temperature of — 459.4° F. suggests 
the absolute zero of thermodynamics. All gases would liquefy or 
even solidify before reaching it. The lowest temperature as yet 
attained is about 450° F. below zero. The absolute zero thus experi- 
mentally conceived (a more strictly absolute scale is discussed later, 
Art. 156) furnishes a convenient starting point for the measure- 
ment of temperature, which will be employed, unless otherwise speci- 
fied, in our remaining discussion. Absolute temperatures are those 
in ivhich the zero point is the absolute zero. Their numerical values 
are to be taken, for the present, at 459.4° greater than those of the cor- 
responding Fahrenheit temperature. 

46. Symbolical Representation. The coefficients determined by Gay-Lussac, 
Charles, and Regnault were those for expansion from an initial volume of 32° F. 
If we take the volume at this temperature as unity, then letting T represent the 
absolute temperature, we have, for the volume at any temperature, 

y- T + 491.4. 
Similarly, for the variation in pressure at constant volume, the initial pressure 
being unity, P —T -=- 491.4. If we let a denote the value 1 -f- 491.4, the first 
expression becomes V — aT, and the second, P = aT. Denoting temperatures on 
the Fahrenheit scale by t, we obtain, for an initial volume v at 32° and any other 
volume V corresponding to the temperature t, produced without change of pressure, 

V=v[l + a(*-32)]. 
Similarly, for variations in pressure at constant volume, 
P=p[l + cr(f-32)]. 



LAWS OF GASES: ABSOLUTE TEMPERATURE 



23 



The value of a is experimentally determined to be very nearly the same for pres- 
sure changes as for volume changes ; the difference in the case of air being less 
than \ of one per cent. The temperature interval between the melting of ice and 
the boiling of water being 180°, the expansion of volume of a gas between those 

limits is — = 0.365, whence Rankine's equation, originally derived from the 

491.4 ^ ° J 

experiments of Regnault and Rudberg, 



PV 



= 1.365, 



in which P, V refer to the higher temperature, and p, v to the lower. 



47. Deviations from Charles' Law. The laws thus enunciated are now known 
not to hold rigidly for any actual gases. For hydrogen, nitrogen, oxygen, air, 
carbon monoxide, methane, nitric oxide, and a few others, the disagreement is 
ordinarily very slight. For carbon dioxide, steam, and ammonia, it is quite pro- 
nounced. The reason for this is that stated in Art. 39. The first four gases named 
have expansive coefficients, not only almost unvarying, but almost exactly identical. 
They may be regarded as our most nearly perfect gases. For air, for example, 
Regnault found over a range of temperature of 180° F., and a range of pressure 
of from 109.72 mm. to 4992.09 mm., an extreme variation in the 
coefficients of only 1.67 per cent. For carbon dioxide, on the 
other hand, with the same range of temperatures and a de- 
creased pressure range 
of from 785.47 mm. 
to 4759.03 mm., the 



y////////w//;/;;;;/w 



variation was 



4.72 



per cent of the lower 
value (2). 



< 



9/////////////////W///////s///////s//////M 



48. The Air Thermometer. The law of Charles sug- 
gests a form of thermometer far more accurate than the 
ordinary mercurial instrument. 
If we allow air to expand with- 
out change in pressure, or to 
increase its pressure without 
change in volume, then we have 
by measurement of the volume 
or of the pressure respectively a 
direct indication of absolute tem- 
perature. The apparatus used 
by Gay-Lussac (Fig. 3), or, 
equally, that shown in Fig. 4, is in fact an air ther- 
mometer, requiring only the establishment of a scale to fit it for practical 
use. A simple modern form of air thermometer is shown in Fig. 6. The 





Fig. 6. 



Art. 48. — Air Ther- 
mometer. 



Fig. 7. Art. 48.— 
Preston Air 
Thermometer. 



24 



APPLIED THERMODYNAMICS 



Fig. 8. Art. 48. 
— H o a cl 1 e y 
Air Ther- 
mometer. 



bulb A contains dry air, and communicates through a tube 
of fine bore with the short arm of the manometer BB, by 
means of which the pressure is measured. The level of the 
mercury is kept constant at a by means of the movable 
reservoir R and flexible tube m. The Preston air ther- 
mometer is shown in Pig. 7. The air is kept at constant 
volume (at the mark a) by admitting mercury from the 
bottle A through the cock B. In the Hoadley air ther- 
mometer, Fig. 8, no attempt is made to keep the volume 
of air constant; expansion into the small tube below the 
bulb increasing the volume so slightly that the error is com- 
puted not to exceed 5° in a range of 600° (3). 

49. Remarks on Air Thermometers. Following Regnault, 
the instrument is usually constructed to measure pressures at 
constant volume, using either nitrogen, hydrogen, or air as a 
medium. Only one " fixed point " need be marked, that of the 
temperature of melting ice. Having marked at 32° the atmos- 
pheric pressure registered at this temperature, the degrees are 
spaced so that one of them denotes an augmentation of pressure 
of 14.7 -=- 491.4 = 0.0299 lb. per square inch. It is usually more 
convenient, however, to determine the two fixed points as usual 
and subdivide the intervening distance into 180 equal degrees. 
The air thermometer readings differ to some extent from those of 
the most accurate mercurial instruments, principally because of 
the fact that mercury expands much less than any gas, and the 
modifying effect of the expansion of the glass container is there- 
fore greater in its case. The air thermometer is itself not a 
perfectly accurate instrument, since air does not exactly follow 
Charles' law (Art. 47). The instrument is used for standardizing 
mercury thermometers, for direct measurements of temperatures 
below the melting point of glass (600-800° F.), as in Regnault's 
experiments on vapors; or, by using porcelain bulbs, for measur- 
ing much higher temperatures. 

50. The Perfect Gas. If actual gases conformed pre- 
cisely to the laws of Boyle and Charles, many of their 
thermal properties might be computed directly. The 
slightness of the deviations which actually occur sug- 
gests the notion of a perfect gas, which would exactly 
and invariably follow the laws, 

PV=c, V P = aT, P v = aT. 

Any deductions which might be made from these sym- 
bolical expressions would of course be rigorously true only 



THE PERFECT GAS 25 

for a perfect gas, which does not exist in nature. T7ie current thermo- 
dynamic method is, however, to investigate the properties of such a gas, modi- 
fying the residts obtained so as to make them applicable to actual gases, 
rather than to undertake to express symbolically or graphically as a 
basis for computation the erratic behavior of those actual gases. The 
error involved in assuming air, hydrogen, and other " permanent " gases 
to be perfect is in all cases too small to be of importance in engineering 
applications. Zeuner (4) has developed an "equation of condition" or 
"characteristic equation" for air which holds even for those extreme con- 
ditions of temperature and pressure which are here eliminated. 

51. Properties of the Perfect Gas. The simplest definition is that 
the perfect gas is one which exactly follows the laws of Boyle and 
Charles. (Rankine's definition (5) makes conformity to Dalton's 
law the criterion of perfectness.) Symbolically, the perfect gas con- 
forms to the law, readily deduced from Art. 50, 

PV=RT, 
in which R is a constant and T the absolute temperature. Consid- 
ering air as perfect, its value for R may be obtained from experi- 
mental data at atmospheric pressure and freezing temperature : 
R = PV+ ^=(14.7 x 144 x 12.387) -h 491.4 = 53.36 foot-pounds. 

For other gases treated as perfect, the value of R may be readily 
calculated when any corresponding specific volumes, pressures, and 
temperatures are known. Under the pressure and temperature just 
assumed, the specific volume of hydrogen is 178.83 ; of nitrogen, 
12.75; of oxygen, 11.20. A useful form of the perfect gas equation 
may be derived from that just given by noting that PF-i- T= R, a 
constant : PV pv 

52. Significance of R. At the standard pressure and temperature 
specified in Art. 51, the values of R for various gases are obviously 
proportional to their specific volumes or inversely proportional to their 
densities. This leads to the form of the characteristic equation some- 
times given, PV = rT -=- M, in which M is the molecular weight and r a 
constant having the same value for all sensibly perfect gases. 

53. Molecular Condition. The perfect gas is one in which the molecules move 
with perfect freedom, the distances between them being so great in comparison 
with their diameters that no mutually attractive forces are exerted. No per- 
formance of disgregation work accompanies changes of pressure or temperature. 



26 APPLIED THERMODYNAMICS 

Hirschfeld (6), in fact, defines the perfect gas as a substance existing in such a 
physical state that its constituent particles exert no interattraction. The coefficient 
of expansion, according to Charles' law, would be the exact reciprocal of the abso- 
lute temperature of melting ice, for all pressures and temperatures. Zeuner has 
shown (7) that as necessary consequences of the theory of perfect gases it can be 
proved that the product of the molecular weight and specific volume, at the same 
pressure and temperature, is constant for all gases ; whence he derives Avogadro's 
principle (Art. 40). Rankine (8) has tabulated the physical properties of the 
" perfect gas." 

54. Kinetic Theory of Gases. Beginning with Bernouilli in 1738, various 
investigators have attempted to explain the phenomena of gases on the basis of 
the kinetic theory, which is now closely allied with the mechanical theory of heat. 
According to the former theory, the molecules of any gas are of equal mass and 
like each other. Those of different gases differ in proportions or structure. The 
intervals between the molecules are relatively very great. Their tendency is to 
move with uniform velocity in straight lines. Upon contact, the direction of mo- 
tion undergoes a change. In any homogeneous gas or mixture of gases, the mean 
energy due to molecular motion is the same at all parts. The pressure of the gas 
per unit of superficial area is proportional to the number of molecules in a unit of 
volume and to the average energy with which they strike this area. It is there- 
fore proportional to the density of the gas and to the average of the squares of the 
molecular velocities. Temperature is proportional to the average kinetic energy 
of the molecules. The more nearly perfect the gas, the more infrequently do the 
molecules collide with one another. When a containing vessel is heated, the mole- 
cules rebound with increased velocity, and the temperature of the gas rises ; when 
the vessel is cooled, the molecular velocity and the temperature are decreased. 
" When a gas is compressed under a piston in a cylinder, the particles of the gas 
rebound from the inwardly moving piston with unchanged velocity relative to 
the piston, but with increased actual velocity, and the temperature of the gas con- 
sequently rises. When a gas is expanded under a receding piston, the particles of 
the gas rebound with -diminished actual velocity, and the temperature falls " (9). 

Recent investigations in molecular physics have led to a new terminology but 
in effect serve to verify and explain the kinetic theory. 

55. Application of the Kinetic Theory. Let w denote the actual molecular 
velocity. Resolve this into components x, y, and z, at right angles to one another. 
Then w 2 - x 2 + y 2 + z 2 . Since the molecules move at random in all directions, 
x — y — 2, and iu 2 = 3 x 2 . Consider a single molecule, moving in an x direction 
back and forth between two limiting surfaces distant from each other d, the x 
component of the velocity of this particle being a. The molecule will make 
(« -^ 2 d) oscillations per second. At each impact the velocity changes from + a 
to — a, or by 2 a, and the momentum by 2 am, if m represents the mass of the 
molecule. The average rate of loss of momentum is 2 am x (a -f- 2 d) = ma 2 h- d; 
and this is the average force exerted per second on the limiting surfaces. The 
total force exerted by all of the molecules on these surfaces is then equal to 

F = ma 2 N = m3^ N = mw^ y in Nvhich N is the tota j nurn ber of molecules in the 
. d d 3d 



THE PERFECT GAS 27 

vessel. Let q be the area of the limiting surface. Then the force per unit of sur- 
face = p = F -=- q = N -f- q = — , whence pv = , in which v is the vol- 

3 d 6 v 3 

ume of the gas = qd (10). 

56. Applications to Perfect Gases. Assuming that the absolute temperature 
is proportional to the average kinetic energy per molecule (Art. 54), this kinetic 
energy being h mw 2 , then letting the mass be unity and denoting by R a constant 
relation, we have pv = RT. The kinetic theory is perfectly consistent with Dal- 
ton's law (Art. 40). It leads also to Avogadro's principle. Let two gases be pres- 
ent. For the first gas, p — nmw 2 -=- 3, and for the second, P — NMW 2 -h- 3. If 
t — T, mw 2 — M IF 2 , and if p = P, then n = N. If M denote the mass of the gas, 
M = mN, and pv = Mw 2 s- 3, or w 2 — 3pv -f- M, from which the mean velocity of 
the molecules may be calculated for any given temperature. 

For gases not perfect, the kinetic theory must take into account, (a) the effect 
of occasional collision of the molecules, and (6) the effect of mutual attractions 
and repulsions. The effect of collisions is to reduce the average distance moved 
between impacts and to increase the frequency of impact and consequently the 
pressure. The result is much as if the volume of the containing vessel were 
smaller by a constant amount, b, than it really is. For v, we may therefore write 
v — b. The value of b depends upon the amount and nature of the gas. The 
effect of mutual attractions is to slow down the molecules as they approach the 
walls. This makes the pressure less than it otherwise w r ould be by an amount 
which can be shown to be inversely proportional to the square of the volume of 
the gas. For p, we therefore write p +(«-f- v 2 ), in which a depends similarly 
ivpon the quantity and nature of the gas. We have then the equation of Van der 
Waals, 

(1) Cf. Verdet, Lemons de Chemie et de Physique, Paris, 1862. (2) Pel. des Exp., 
I, 111, 112. (3) Trans. A. 8. M. E., VI, 282. (4) Technical Thermodynamics 
(Klein tr.), II, 313. (5) " A perfect gas is a substance in such a condition that the 
total pressure exerted by any number of portions of it, against the sides of a vessel in 
which they are inclosed, is the sum of the pressures which each such portion would 
exert if enclosed in the vessel separately at the same temperature.' 1 — The Steam 
Engine, 14th ed., p. 220. (0) Engineering Thermodynamics, 1907. (7) Op. cit., I, 
104-107.- (8) Op. cit., 593-595. (9) Nichols and Franklin, The Elements of Physics, 
I, 199-200. (10) Ibid., 199 ; Wormell, Thermodynamics, 157-161. (11) Over de 
Continuiteit van den Gas en Vloeistoestand, Leinden, 1873, 76 ; tr. by Roth, Leipsic, 
1887. 

SYNOPSIS OF CHAPTER III 

Boyle's law, pv = PV: deviations. 

Dalton's law, Avogadro's principle. 

Law of Gay-Lussac or of Charles : increase of volume at constant pressure ; increase 

of pressure at constant volume ; values of the coefficient from 32° F. ; deviations 

with actual gases. 



28 APPLIED THERMODYNAMICS 

The absolute zero : — 459.4° F., or 491. 4° F. below the freezing point. 
Air thermometers : Preston's : Hoadley's : calibration : gases used. 

The perfect gas, -^ = : definitions : properties : values of B : absence of inter- 
molecular action ; the kinetic theory ; development of the law PV = BT there- 
from ; conformity with Avogadro's principle ; molecular velocity. 

The Van der Waals equation for imperfect gases : 



[p + a -^-V = BT ' 



PROBLEMS 

1. Find the volume of one pound of air at a pressure of 100 lb. per square inch, 
the temperature being 32° F., using Boyle's law only. 

2. From Charles' law, find the volume of one pound of air at atmospheric pres- 
sure and 72° F. 

3. Find the pressure exerted by one pound of air having a volume of 10 cubic 
feet at 32° F. 

4. One pound of air is cooled from atmospheric pressure at constant volume from 
32° F. to — 290° F. How nearly perfect is the vacuum produced ? 

5. Air at 50 lb. per square inch pressure at the freezing point is heated at con- 
stant volume until the temperature becomes 2900° F. Find its pressure after heating. 

6. Five pounds of air occupy 50 cubic feet at a temperature of 0° F. Find the 
pressure. 

7. Find values of B for hydrogen, nitrogen, oxygen. 

8. Find the volume of three pounds of hydrogen at 15 lb. pressure per square 
inch and 75° F. 

9. Find the temperature of 2 ounces of hydrogen contained in a 1 -gallon flask 
and exerting a pressure of 10,000 lb. per square inch. 

10. Compute the value of r (Art. 52) . 

11. Find the mean molecular velocity of 1 lb. of air (considered as a perfect gas) 
at atmospheric pressure and 70° F. 

12. How large a flask will contain 1 lb. of nitrogen at 3200 lb. pressure per 
square inch and 70° F. ? 



CHAPTER IV 

THERMAL CAPACITIES : SPECIFIC HEATS OF GASES : JOULE'S LAW 

57. Thermal Capacity. The definition of specific heat given in Art. 24 is, 
from a thermodynamic standpoint, inadequate. Heat produces other effects than 
change of temperature. A definite movement of heat can be estimated only when 
all of these effects are defined. For example, the quantity of heat necessary to 
raise the temperature of air one degree in a constant volume air thermometer is 
much less than that used in raising the temperature one degree in the constant 
pressure type. The specific heat may be satisfactorily defined only by referring 
to the condition of the substance during the change of temperature. Ordinary 
specific heats assume constancy of pressure, — that of the atmosphere, — while the 
volume increases with the temperature in a ratio which is determined by the coeffi- 
cient of expansion of the material. A specific heat determined in this way — as 
are those of solids and liquids generally — is the specific heat at constant pressure. 

Whenever the term " specific heat" is used without qualification, this par- 
ticular specific heat is intended. Heat may be absorbed during changes of 
either pressure, volume, or temperature, while some other of these proper- 
ties of the substance is kept constant. For a specific change of property, 
the amount of heat absorbed represents a specific thermal capacity. 

58. Expressions for Thermal Capacities. If H represents heat absorbed, 
c a constant specific heat, and (T — t) a range of temperature, then, by 
definition, H=c(T— t) and c = H+(T—t). If c be variable, then 

H= J cdT and c — dH^dT. If in place of c we wish to denote the 

specific heat at constant pressure (k), or that at constant volume (I), we may 
apply subscripts to the differential coefficients ; thus, 

jc=(<m ^i=m 

\dTJ P \dT 

the subscripts denoting the property which remains constant during the 
change in temperature. 



We have also the thermal capacities, 

fd_H\ fdH\ /dH\ (dH\. 
WK/j' \dP It \dPJr \dVJp 



The first of these denotes the amount of heat necessary to increase the specific 
volume of the substance by unit volume, while the temperature remains constant ; 

29 



30 APPLIED THERMODYNAMICS 

this is known as the latent heat of expansion. It exemplifies absorption of heat 
without change of temperature. No names have been assigned for the other 
thermal capacities, but it is not difficult to describe their significance. 

59. Values of Specific Heats. It was announced by Dulong and Petit that the 
specific heats of substances are inversely as their chemical equivalents. This was 
shown later by the experiments of Regnault and others to be approximately, 
though not absolutely, correct. Considering metals in the solid state, the product 
of the specific heat by the atomic weight ranges at ordinary temperatures from 6.1 
to 6.5. This nearly constant product is called the atomic heat. Determination of 
the specific heat of a solid metal, therefore, permits of the approximate computa- 
tion of its atomic weight. Certain non-metallic substances, including chlorine, 
bromine, iodine, selenium, tellurium, and arsenic, have the same atomic heat as 
the metals. The molecular heats of compound bodies are equal to the sums of the 
atomic heats of their elements; thus, for example, for common salt, the specific 
heat 0.219, multiplied by the molecular weight, 58.5, gives 12.8 as the molecular 
heat ; which, divided by 2, gives 6.4 as the average atomic heat of sodium and 
chlorine; and as the atomic heat of sodium is known to be 6.4, that of chlorine 
must also be 6.4 (1). 

60. Volumetric Specific Heat. Since the specific volumes of gases are in- 
versely as their molecular weights, it follows from Art. 59 that the quotient of the 
specific heat by the specific volume is practically constant for ordinary gases. In 
other words, the specific heats of equal volumes are equal. The specific heats of 
these gases are directly proportional to their specific volumes and inversely pro- 
portional to their densities, approximately. Hydrogen must obviously possess the 
highest specific heat of any of the gases. 

61. Mean, "Real," and " Apparent" Specific Heats. Since all specific 
beats are variable, the values given in tables are mean values ascertained 
over a definite range of temperature. The mean specific heat, adopting 
the notation of Art. 58, is c = H -t-(T-t); while the true specific heat, or 
specific heat "at a point," is the limiting value c — clH-r- dT. 

Rankine discusses a distinction between the real and apparent specific heats; 
meaning by the former, the rate of heat absorption necessary to effect changes of 
temperature alone, without the performance of any disgregation or external work . 
and by the latter, the observed rate of heat absorption, effecting the same change 
of temperature, but simultaneously causing other effects as well. For example, 
in heating water at constant pressure from 62° to 63° F., the apparent specific heat 
is 1.0 (definition, Art. 22). To compute the real specific heat, we must know the 
external work done by reason of expansion against the constant pressure, and the 
disgregation work which has readjusted the molecules. Deducting from 1.0 
the heat equivalent to these two amounts of work, we have the real specific heat, 
that which is used solely for making the substance hotter. Specific heats determined 
by experiment are always apparent; the real specific heats are known only by 
computation (Art. 64). 



SPECIFIC HEATS OF GASES 31 

62. Specific Heats of Gases. Two thermal capacities of especial 
importance are used in calculations relating to gases. The first is 
the specific heat at constant pressure, k, which is the amount of heat 
necessary to raise the temperature one degree while the pressure is kept 
constant; the other, the specific heat at constant volume, 1, or the 
amount of heat necessary to raise the temperature one degree while the 
volume 'is kept constant. 

63- Regnault's Law. As a result of his experiments On a large number of 
gases over rather limited ranges of temperature, Regnault announced that the 
specific heat of any gas at constant pressure is constant. This is now known not to 
be rigorously true of even our most nearly perfect gases. It is not even approxi- 
mately true of those gases when far from the condition of perfectness, i.e. at low 
temperatures or high pressures. At very Ugh temperatures, also, it is well known 
that specific heats rapidly increase. This particular variation is perhaps due to 
an approach toward that change of state described as dissociation. When near 
any change of state, — combustion, fusion, evaporation, dissociation, — every sub- 
stance manifests erratic thermal properties. The specific heat of carbon dioxide 
is a conspicuous illustration. Recent determinations by Holborn and Henniug^ 
(2) of the mean specific heats between 0° and x° C. givej, for nitrogen, k = 0.255 
+ 0.000019 a;; and for carbon dioxide, k = 0.201 + 0.0000742 x - 0.000000018 z 2 ; 
while for steam, heated from 110° to x° C, k = 0.4669 -0.0000168 x + 0.000000044 x\ 
The specific heats of solids also vary. The specific heats of substances in general 
increase with the temperature. Regnault's law would hold, however, for a perfect 
gas; in this, the specific heat would be constant under all conditions of tempera- 
ture. For our "permanent" gases, the specific heat is practically constant at, 
ordinary temperatures. ■ j 

■ t't 
64. The Two Specific Heats. When a gas is heated at constant pressure/ 
its volume increases against that pressure, and external work is done in 
consequence. The external work may be computed by multiplying the 
pressure by the change in volume. When heated at constant volume, no 
external work is done; no movement is made against an external resist- 
ance. ' If the gas be perfect, then, under this condition, no disgregation 
work is done ; and the specific heat at constant volume is a true specific 
heat, according to Rankine's distinction (Art. 61). The specific heat at 
constant pressure is, however, the one commonly determined by experi- 
ment. The numerical values of the two specific heats must, in a perfect 
gas, differ by the heat equivalent to the external work done during heating 
at constant pressure. Under certain conditions, — as with water at its 
maximum density, — no external work is done when heating at constant 
pressure ; and at this state the two specific heats are equal, if we ignore 
possible differences in the disgregation work. 



32 APPLIED THERMODYNAMICS 

65. Difference of Specific Heats. Let a pound of air at 32° F. 

and atmospheric pressure be raised 1° F. in temperature, at constant 
pressure. It will expand 12.387 -j- 491.4 = 0.02521 cu. ft., against 
a resistance of 14.7 x 144 = 2116. 8 lb. per square foot. The external 
work which it performs is consequently 2116.8 x 0.02521 = 53.36 foot- 
pounds. A general expression for this external work is W= P F-s- T\ 
and as from Art. 51 the quotient PT-f ^is a constant and equal to 
i2, then IT is a constant for each particular gas, and equivalent in 
value to that of R for such gas. The value of W for air, expressed 
in heat units, is 53.36-j-778 = 0.0686. If the specific heat of air at 
constant pressure, as experimentally determined, be taken at 0.2375, 
then the specific heat at constant volume is 0.2375 — 0.0686 = 0.1689, 
air being regarded as a perfect gas. 

66. Derivation of Law of Perfect Gas. Let a gas expand at constant pres- 
sure P from the condition of absolute zero to any other condition V, T. The total 
external work which it will have done in consequence of this expansion is PV. 
The work done per degree of temperature is PV-- T. But, by Charles' law, this 
is constant, whence we have PV=RT. The symbol R of Art. 51 thus represents 
the external work of expansion during each degree of temperature increase (3). 

67. General Case. The difference of the specific heats, while constant for any 
gas, is different for different gases, because their values of R differ. But since 
values of R are proportional to the specific volumes of gases (Art. 52), the differ- 
ence of the volumetric specific heats is constant for all gases. Thus, let k, I be the 
two specific heats, per pound, of air. Then k — I = r. Let d be the density of 
the air; then, d(k—l) is the difference of the volumetric specific heats. For any 
other gas, we have similarly, K — L — R and D(K — L) ; but, from Art. 52 
R:r::d:D, or R = rd - D. Hence, K - L = rd - D = (k - l)(d h- D), or 
D(K — IS)— d(k — I). The difference of the volumetric specific heats is for all 
gases equal approximately to 0.0055 B. t. u. (Compare Art. 60.) 

68. Computation of External Work. The value of R given in Art. 52 and 
Art. 65 is variously stated by the writers on the subject, on account of the 
slight uncertainty which exists regarding the exact values of some of the con- 
stants used in computing it. The differences are too small to be of consequence 
in engineering work. 

69. Ratio of Specific Heats. The numerical ratio between the 
two specific heats of a sensibly perfect gas, denoted by the symbol y, 
is a constant of prime importance in thermodynamics. 

For air, its value is 0.2375 -=- 0.1689 = 1.402. Various writers, using other 
fundamental data, give slightly different values (4). The best direct experiments 



SPECIFIC HEATS OF GASES 33 

(to be described later) agree with that here given within a narrow margin. For 
hydrogen, Lnmmer and Pringsheim (5) have obtained the value 1.408; and for 
oxygen, 1.396. For carbon dioxide, a much less perfect gas than any of these, 
these observers make the value of y, 1.2961 ; while Dulong obtained 1.338. The 
latter obtained for carbon monoxide 1.428. The mean value for the " permanent " 
gases is close to that for air, viz., 

y = 1.402. ; 

The value should be the same for all gases as they closely approach 
perfectness ; for as the law PV= RT holds, so must the difference of 
the specific heats be absolutely constant ; and as Regnault's law (Art. 63) 
holds, the two specific heats must themselves be constant ; whence their 
ratio must also be constant. The value of the ratio for ordinary actual 
gases is independent of the temperature and the pressure. 

70. Relations of R and y. A direct series of relations exists 
between the two specific heats, their ratio, and their difference. If 
we denote the specific heats by k and Z, then in proper units, 

k — I = R. l = h — R. -t=V' -z ~ = V - 

I u k-R u 



(W air, this gives 0-2375 = 1.402.^) 

V 6 , no , t 53.36 J 



0.2375 

k = ky — yR. ky—k — yR. k — R—^—r- 

l=R— — R = k^^ = l(y-1). 

y~ l y 

71. Ranklne's Prediction of the Specific Heat of Air. The specific heat of air 
was approximately determined by Joule in 1852. Earlier determinations were 
unreliable. Rankine, in 1850, by the use of the relations just cited, closely ap- 
proximated the result obtained experimentally by Regnault three years later. 
Using the values y = 1.4, R = 53.15, Rankine obtained 

h = R -J— = (53.15 h- 772) x (1.4 - 0.4) = 0.239. 

y- i 

Regnault's result was 0.2375. 

72. Mayer's Computation of the Mechanical Equivalent of Heat. 

Reference was made in Art. 29 to the computation of this constant 
prior to the date of Joule's conclusive experiments. The method is 
substantially as follows : a cylinder and piston having an area of one 
square foot, the former containing one cubic foot, are assumed to hold 



34 



APPLIED THERMODYNAMICS 



air at 32° F., which is subjected to heat. The piston is balanced, so 
that the pressure on the air is that of the atmosphere, or 14.7 lb. 
per square inch ; the total pressure on the piston being, then, 
144 x 14.7 = 2116.8 lb. While under this pressure, the air is heated 
until its temperature has increased 491.4°. The initial volume 
of the air was by assumption one cubic foot, whence its weight 
was 1 -r- 12.387 = 0.0811 lb. The heat imparted was therefore 
0.0811 x 0.2375 x 491.4 = 9.465 B. t. u. The external work was 
that due to doubling the volume of the air, or 1 x 14.7 x 144 = 2116.8 
foot-pounds. The piston is now fixed rigidly in its original position, 
so that the volume cannot change, and no external work can be done. 
The heat required to produce an elevation of temperature of 491.4° 
is then 0.0811x0.1689x491.4 = 6.731 B. t, u. the difference 
of heat corresponding to the external work done is 2.734 B. t. u., 
whence the mechanical equivalent of heat is 2116.8 -r- 2.734= 774.2 
foot-pounds. 

73. Joule's Experiment. One of the crucial experiments of the science was 
conducted by Joule about 1844, after having been previously attempted by Gay- 
Lussac. 

Two copper receivers, A and B, Fig. 9, were connected by a tube 
and stopcock, and placed in a water bath. Air was compressed in A 

to a pressure of 22 atmospheres, 
while a vacuum was maintained 
in B. When the stopcock was 
opened, the pressure in each re- 
ceiver became 11 atmospheres, and 
the temperature of the air and of 
the water bath remained practically- 
unchanged. This was an instance of expansion without the perform- 
ance of external work ; for there was no resisting pressure against the 
augmentation of volume of the air. 




Fig. 9. Arts. 73, 80. — Joule's Experiment. 



74. Joule's and Kelvin's Porous Plug Experiment. Minute observations 
showed that a slight change of temperature occurred in the water bath. 
Joule and Kelvin, in 1852, by their celebrated "porous plug" experiments, 
ascertained the exact amount of this change for various gases. In all of 
the permanent gases the change was very small ; in some cases the tern- 



JOULE'S LAW 35 

perature increased, while in others it decreased ; and the inference is jus- 
tified that in a perfect gas there would be no change of temperature (Art. 
156). 

75. Joule's Law. The experiments led to the principle that 
when a perfect gas expands without doing external work, and without 
receiving or discharging heat, the temperature remains unchanged and 
no disgregation work is done. A clear appreciation of this law is of 
fundamental importance. Four thermal phenomena might have 
occurred in Joule's experiment : a movement of heat, the performance 
of external work, a change in temperature, or work of disgregation. 
From Art. 12, these four effects are related to one another in such 
manner that their summation is zero; (H= T+ 1+ W). By means 
of the water bath, which throughout the experiment had the same 
temperature as the air, the movement of heat to or from the air was 
prevented. By expanding into a vacuum, the performance of external 
work was prevented. The two remaining items must then sum up 
to zero, i.e. the temperature change and the disgregation work. But 
the temperature did not change ; consequently the amount of disgre- 
gation work must have been zero. 

76. Consequences of Joule's Law. In the experiment described, the pres- 
sure and volume changed without changing the internal energy. No dis- 
gregation work was done, and the temperature remained unchanged. 
Considering pressure, volume, and temperature as three cardinal thermal 
properties, internal energy is then independent of the pressure or volume 
and depends on the temperature only, in any perfect gas. It is thus itself 
a cardinal property, in this case, a function of the temperature. "A 
change of pressure and volume of a perfect gas not associated with change 
of temperature does not alter the internal energy. In any change of tem- 
perature, the change of internal energy is independent of the relation of 
pressure to volume during the operation ; it depends only on the amount 
by which the temperature has been changed" (6). The gas tends to cool 
in expanding, but this effect is " exactly compensated by the heat which 
is disengaged through the friction in the connecting tube and the im- 
pacts which destroy the velocities communicated to the particles of gas 
while it is expanding" (7). There is practically no disgregation work in 
heating a sensibly perfect gas; all of the internal energy is evidenced by 
temperature alone. When such a gas passes from one s,tate to another in 
a variety of ways, the external work done varies; but if from the total 



36 APPLIED THERMODYNAMICS 

movement of heat the equivalent of the external work be deducted, then 
the remainder is always the same, no matter in what way the change of 
condition has been produced. Instead of H = T + I -f- W, we may write 
H= T+ W. 

77. Application to Difference of Specific Heats. The heat absorbed dur- 
ing a change in temperature at constant pressure being H=k(T—t), and 
the external work during such a change being W= P(V— v) — R(T—t), 
the gain of internal energy must be 

H- W=(k-r)(T-t). 

The heat absorbed during the same change of temperature at constant 
volume is H=l(T—t). Since in this case no external work is done, the 
whole of the heat must have been applied to increasing the internal energy. 
But, according to Joule's law, the change of internal energy is shown by the 
temperature change alone. In whatever way the temperature is changed 
from T to t, the gain of internal energy is the same. Consequently, 

(k- R)(T-t)=l(T -t) and k-R = l, 

a result already suggested in Art. 65. 

78. Discussion of Results. The greater value of the specific heat at 
constant pressure is due solely to the performance of external work dur- 
ing the change in temperature. The specific heat at constant volume is 
a real specific heat, in the case of a perfect gas ; no external work is done, 
and the internal energy is increased only by reason of an elevation of tem- 
perature. There is no disgregation work. All of the heat goes to make 
the substance hot. So long as no external work is done, it is not neces- 
sary to keep the gas at constant volume in order to confirm the lower 
value for the specific heat; no more heat is required to raise the tempera- 
ture a given amount when the gas is allowed to expand than when the 
volume is maintained constant. For any gas in which the specific heat at 
constant volume is constant, Joule's law is inductively established ; for no 
external work is done, and temperature alone measures the heat absorp- 
tion at any point on the thermometric scale. If a gas is allowed to expand, 
doing external work at constant temperature, then, since no change of inter- 
nal energy occurs, it is obvious from Art. 12 that the external work is equal 
to the heat absorbed. Briefly, the important deduction from Joule's experi- 
ment is that item (b), Art. 12, may be ignored when dealing with sensibly 
perfect gases. 

79. Confirmatory Experiment. By a subsequent experiment, Joule 
showed that when a gas expands so as to perform external work, heat dis- 




JOULE'S LAW 37 

appears to an extent proportional to the work done. Figure 10 illustrates 
the apparatus. A receiver A, containing gas compressed to two atmos- 
pheres, was placed in the calorimeter B and connected with the gas holder 
C, placed over a water tank. The gas passed 
from A to C through the coil D, depressed the 
water in the gas holder, and divided itself be- 
tween the two vessels, the pressure falling to 
that of one atmosphere. The work done was 

computed from the augmentation of volume shown FlG - 10 - Art - '9. —Joule's 
, t . . , , ~ . Experiment, Second Ap- 

by driving down the water m C against atmos- . para t U s. 

pheric pressure ; and the heat lost was ascertained 

from the fall of temperature of the water. If the temperature of the 

air were caused to remain constant throughout the experiment, then the 

work done at G would be precisely equivalent to the heat given up by 

the water. If the temperature of the air were caused to remain constantly 

the same as that of the water, then H= = T-f- /+ W, (T + I) = - W, or 

internal energy would be given up by the air, precisely equivalent in amount 

to the work done in C. 

80- Application of the Kinetic Theory. In the porous plug experiment referred 
to in Art. 71, it was found that certain gases were slightly cooled as a result of the 
expansion, and others slightly warmed. The molecules of gas are very much closer 
to one another in A than in B, at the beginning of the experiment. If the mole- 
cules are mutually attractive, the following action takes place : as they emerge from 
A, they are attracted by the remaining particles in that vessel, and their velocity 
decreases. As they enter B, they encounter attractions there, which tend to in- 
crease their velocity; but as the second set of attractions is feebler, the total effect 
is a loss of velocity and a cooling of the gas. In another gas, in which the molecules 
repel one another, the velocity during passage would be on the whole augmented, 
and the temperature increased. A perfect gas would undergo neither increase nor 
decrease of temperature, for there would be no attractions or repulsions between 
the molecules. 

(1) A critical review of this theory has been presented by Mills : The Specific 
Heats of the Elements, Science, Aug. 24, 1908, p. 221. (2) The Engineer, January 
13, 1908. (3) Throughout this study, no attention will be paid to the ratio 778 as 
affecting the numerical value of constants in formulas involving both heat and work 
quantities. The student should discern whether heat units or foot-pounds are in- 
tended. (4) Zeuner, Technical Thermodynamics, Klein tr., I, 121. (5) Ibid., loc. 
cit. (6) Ewing : The Steam Engine, 1906. (7) Wormell, Thermodynamics. 



SYNOPSIS OF CHAPTER IV 

Specific thermal capacities ; at constant pressure, at constant volume : other capacities. 

Atomic heat = specific heat x atomic weight ; molecular heat. 

The volumetric specific heats of common gases are approximately equal. 



38 APPLIED THERMODYNAMICS 

TT fJ TT 

Mean specific heat = ; true specific heat = — ; real and apparent specific heats. 

Begnaulfs law : " the specific heat is constant for perfect gases." 

Difference of the two specific heats : B = 53.36 ; significance of B. 

The difference of the volumetric specific heats equals 0.0055 B. t. u. for all gases. 

Ratio of the specific heats : y = 1.402 for air ; relations between k, Z, ?/, B. 

Rankine's prediction of the value of k : Mayer's computation of the mechanical equiva- 
lent of heat. 

Joule' 1 's Law : no disgregation work occurs in a perfect gas. 

If the temperature does not change, the external work equals the heat absorbed. 

If no heat is received, internal energy disappears . to an extent equivalent to the 
external work done. 

The condition of intermolecular force determines whether a rise or a fall of temperature 
occurs in the porous plug experiment. 



PROBLEMS 

1. The atomic weights of iron, lead, and zinc being respectively 56, 206.4, 65; and 
the specific heats being, for cast iron, 0.1298 ; for wrought iron, 0.1138 ; for lead, 
0.0314 ; and for zinc, 0.0956, — check the theory of Art. 59 and comment on the results. 

2. Find the volumetric specific heats at constant pressure of air, hydrogen, and 
nitrogen, and compare with Art. 60. (k = 3.4 for H and 0.2438 for N.) 

3. The heat expended in warming water from 32° F. to 160° F. being 127.86 B. t. u. , 
find the mean specific heat over this range. 

4. The weight of a cubic foot of water being 59.83 lb. at 212° F. and 62.422 lb. at 
32° F., find the amount of heat expended in performing external work when one pound 
of water is heated between these temperatures at atmospheric pressure. 

5. (a) Find the specific heat at constant volume of hydrogen and nitrogen. 
(b) Find the value of y for these two gases. 

6. Check the value 0.0055 B. t. u. given in Art. 67 for hydrogen and nitrogen. 

7. Compute the elevation in temperature, in Art. 72, that would, for an expansion 
of 100 per cent, under the assumed conditions, and with the given values of k and I, 
give 'exactly 778 as the value of the mechanical equivalent of heat. What law of 
gaseous expansion would be invalidated if this elevation of temperature occurred ? 

8. In the experiment of Art. 79, the volume of air in C increased by one cubic foot 
against normal atmospheric pressure. The weight of water in B was 20 lb. The tem- 
perature of the air remained constant throughout the experiment. Ignoring radiation 
losses, compute the fall of temperature of the water. 

9. Prove that the specific heat at constant pressure is constant for a perfect gas. 



CHAPTER V . 

GRAPHICAL REPRESENTATIONS: PRESSURE-VOLUME PATHS OF 

PERFECT GASES 

81. Thermodynamic Coordinates. The condition of a body being fully 
defined by its pressure, volume, and temperature, its state may be repre- 
sented on a geometrical diagram in which these properties are used as 
coordinates. This graphical method of analysis, developed by Clapeyron, 
is now in universal use. The necessity for three coordinates presupposes 
the use of analytical geometry of three dimensions, and representations 
may then be shown perspectively as related to one of the eight corners 
of a cube; but the projections on any of the three adjacent cube faces are 
commonly used ; and since any two of three properties fix the third when 
the characteristic equation is known, a projective representation is suffi- 
cient. Since internal energy is a cardinal property (Arts. 10, 76), this also 
may be employed as one of the coordinates of a diagram if desired. 

82. Illustration. In Fig. 11 we have one corner of a cube 
constituting an origin of coordinates at O. The temperature of a 
substance is to be represented by the distance upward from 0; its 
pressure, by the distance to the right ; and its volume, by the dis- 
tance to the left. The lines forming the cube edges are correspond- 
ingly marked OT, OP, OV. Consider the condition of the body to 
be represented by the point A, within the cube. Its temperature is 
then represented by the distance AB, parallel to TO, the point B 
being in the plane VOP. The distance AD, parallel to P 0, from A 
to the plane TOV, indicates the pressure ; and by drawing AC paral- 
lel to VO, being the intersection of this line with, the plane TOP, 
we may represent the volume. The state of the substance is thus 
fully shown. Any of the three projections, Figs. 12-14, would equally 
fix its condition, providing the relation between P, V, and T is 
known. In each of these projections, two of the properties of the 
substance are shown ; in the three projections, each property appears 

39 



40 



APPLIED THERMODYNAMICS 



twice; and the corresponding lines AB, AC, and AD are always 
equal in length. 




A,C, 



A,B, 



A.D, 



Fig. 11. Art. 82. — 
Perspective Dia- 
gram. 



Fig. 12. Art. 82.— 
TP Diagram. 



v, — c r v B 

Fig. 13. Art. 82.— Fig. 14. Art. 82. 
VP Diagram. TV Diagram. 



83. Thermal Lines. In Fig. 15, let a substance, originally at A, pass 
at constant pressure and temperature to the state B\ thence at constant 
temperature and volume to the state C; and thence at constant pressure 




o 
Fig. 15. Art. 83.— 
Perspective Ther- 
mal Line. 



AB. 



-P 



C,D, 



B,C, 



Fig. 16. Art. 83. 
TP Path. 



Fig. 17. Art. 83. 
VP Path. 



Fig. 18. Art. 83. 
TV Path. 



and volume to D. Its changes are represented by the broken line ABCD, 
which is shown in its various projections in Figs. 16-18. The thermal 
line of the coordinate diagrams, Figs. 11 and 15, is the locus of a series of 
successive states of the substance. A path is the projection of a thermal 
line on one of the coordinate planes (Figs. 12-14, 16-18). The path of a 
substance is sometimes called its process curve, and its thermal line, a 
thermogram. 

The following thermal lines are more or less commonly studied : — 

(a) Isothermal, in which the temperature is constant; its plane is 

perpendicular to the O^axis. 

(b) Isometric, in which the volume is constant ; having its plane per- 

pendicular to the OF' axis. 

(V) Isopiestic, in which the pressure is constant ; its plane being per- 
pendicular to the OP axis. 

(of) Isodynamic, that along which no change of internal energy 
occurs. 



GRAPHICAL REPRESENTATIONS 



41 



(e) Adiabatic, that along which no heat is transferred between the 
substance and surrounding bodies; the thermal line of an 
insulated body. 

84. Thermodynamic Surface. Since the equation of a gas in- 
cludes three variables, its geometrical representation is a surface ; 
and the first three, at least, of the above paths, must be projections 
of the intersection of a plane with such surface. Figure 19, from Pea- 




Arts. 84, 103.— Thermodynamic Surface for a Perfect Gas. 



body (1), admirably illustrates the equation of a perfect gas, PV ' — 
RT. The surface pmnv is the characteristic surface for a perfect gas. 
Every section of this surface parallel to the PF'plane is an equilat- 
eral hyperbola. Every projection of such section on the PV plane 
is also an equilateral hyperbola, the coordinates of which express the 
law of Boyle, PV—C. Every section parallel with the TV plane 
gives straight lines pm, sZ, etc., and every section parallel with the 
TP plane gives straight lines vn, xy, etc. The equations of these 



42 APPLIED THERMODYNAMICS 

lines are expressions of the two forms of the law of Charles, their 
appearance being comparable with that in Fig. 5. 

85. Path of Water at Constant Pressure. Some such diagram as that 
of Fig. 20 would represent the behavior of water in its solid, liquid, and 

vaporous forms when heated at constant pressure. 

The coordinates are temperature and volume. At 

A, the substance is ice, at a temperature below 

the freezing point. As the ice is heated from A 

to B, it undergoes a slight expansion, like other 

D '\ solids. At B, the melting point is reached, and 

as ice contracts in melting, there is a decrease in 

volume at constant temperature. At C, the sub- 

-r. ™ * j. ^ ™ stance is all water: it contracts until it reaches the 
Fig. 20. Art. 85. — Water ' 

at Constant Pressure. temperature of maximum density, 39.1° F., at D, 

then expands until it boils at E, when the great 
increase in volume of steam over water is shown by the line EF. If the 
steam after formation conformed to Charles' law, the path would con- 
tinue upward and to the right from F, as a straight line. 

86. The Diagram of Energy. Of the three coordinate planes, the PV 
is most commonly used. This gives a diagram corresponding with that 
produced by the steam engine indicator (Art. 484). It is sometimes called 
Watts' diagram. Its importance arises principally from the fact that it 
represents directly the external work done during the movement of the 
substance along any path. Consider a vertical cylinder filled with fluid, 
at the upper end of which is placed a weighted piston. Let the piston be 
caused to rise by the expansion of the fluid. The force exerted is then 
equivalent to the weight of the piston, or total pressure on the fluid ; the 
distance moved is the movement of the piston, which is equal to the aug- 
mentation in volume of the fluid. Since work equals force multiplied by 
distance moved, the external work done is equal to the total uniform pressure 
multiplied by the increase of volume. 

87. Theorem. On a PV diagram, the external work done along 
any path is represented by the area included be- 
tween that path and the perpendiculars from its 
extremities to the horizontal axis. 

Consider first a path of constant pressure, ah, 
Fig. 21. From Art. 86, the external work is 
equivalent to the pressure multiplied by the in- FlG - 21 - Art - 87 — 

^ L sy External Work at 

crease of volume, or to ea x ab = cabd. G-eneral Constant Pressure. 



d 



CYCLES 



43 



case : let the path be arbitrary, ab, Fig. 22. 
into an infinite number of vertical strips, amnc, 
each of which may be regarded as a rectangle, 
such that ac = mn, mn = op, etc. The external 
work done along am, mo, oq, etc., is then repre- 
sented by the areas amnc, mopn, oqrp, etc., and 
the total external work along the path ab is repre- 
sented by the sum of these areas, or by abdc. 

Corollary I. Along a path of constant volume 
no external work is done. 



Divide the area abdc 
mopn, oqrp, etc., 




c n p r d 

Fig. 22. Arts. 87,88. 
— External Work, 
Any Path. 



Corollary II If the path be reversed, i.e. from right to left, as 
along ba, the volume is diminished, and negative work is done ; work 
is expended on the substance in compressing it, instead of being per- 
formed by it. 



88. Significance of Path. It is obvious, from Fig. 22, that the amount 
of external work done depends not only on the initial and final states a and 
b, but also on the nature of the path between those states. According to 
Joule's principle (Art. 75) the change of internal energy (T+I, Art. 12) 
between two states of a perfect gas is dependent upon the initial and final 
temperatures only and is independent of the path. The external work 
done, however, depends upon the path. The total expenditure of heat, which 
includes both effects, can only be known when the path is given. The 
internal energy of a perfect gas (and, as will presently be shown, Art. 
109, of any substance) is a cardinal property ; external work and heat 
transferred are not. They cannot be used as elements of a coordinate 
diagram. 



89. Cycle. 



°0> 



A series of paths forming a closed finite figure con- 
stitutes a cycle. In a cycle, the substance is brought 
back to its initial conditions of pressure, volume, 
and temperature. 

Theorem. In a cycle, the net external work 
done is represented on the PV diagram by the en- 
closed at ea. 

Let abed, Fig. 23, be any cycle. Along abc, the 
work done is, from Art. 87, represented by the 
area abcef. Along cda, the negative work done is similarly repre- 



Fig. 23. Art. 89.— 
External Work in 
Closed Cycle. 



44 APPLIED THERMODYNAMICS 

sented by the area adcef. The net positive work done is equivalent 
to the difference of these two areas, or to abed. 

If the volume units are in cubic feet, and the pressure units are pounds 
per square foot, then the measured area abed gives the work in foot-pounds: 
This principle underlies the calculation of the horse power of an engine 
from its indicator diagram. If the cycle be worked in a negative direction, 
e.g. as chad, Fig. 23, then the net work will be negative ; i.e. work will 
have been expended upon the substance, adding heat to it, as in an air 
compressor. 

90. Theorem. In a perfect gas cycle, the expenditure of heat is 
equivalent to the external work done. 

Since the substance has been brought back to its initial tempera- 
ture, and since the internal energy depends solely upon the tempera- 
ture, the only heat effect is the external work. In the equation 
H= T+I+ W, T+I= 0, whence H= W, the expenditure of heat 
being equivalent to its sole effect.* 

If the work is measured in foot-pounds, the heat expended is calcu- 
lated by dividing by 778. (See Note 3, page 37.) Conversely, in a 
reversed cycle, the expenditure of external work is equivalent to the gain of 
heat. 

91. Isothermal Expansion. The isothermal path is one of much 
importance in establishing fundamental principles. By definition 
(Art. 83) it is that path along which the temperature of the fluid 
is constant. For gases, therefore, from the characteristic equation, 
if T be made constant, the isothermal equation is 

PV=BT=C. 

Taking R at 53.36 and T at 491.4° (32° F.), C= 53.36 x 491.4 = 
26,221.104 ; whence we plot on Fig. 2 the isothermal curve ab for 
this temperature ; an equilateral hyperbola, asymptotic to the axes 
of P and V. An infinite number of isothermals might be plotted, 
depending upon the temperature assigned, as cd, ef, gh, etc. The 
equation of the isothermal may be regarded as a special form of the 
exponential equation PV n = C, in which n = 1. 

* It may be inferred later (Art. 109) that this theorem is valid for substances in 
general. 



ISOTHERMAL EXPANSION 



45 



92. Graphical Method. For rapidly plotting curves of the form PV = C, the 

construction shown in Fig. 24 is useful. Knowing the three corresponding prop- 
erties of the gas at any given 

state enables us to fix one point 

on the curve ; thus the volume 

12.387 and the pressure 2116.8 

give us the point C on the 

isothermal for 491.4° absolute. 

Through C draw CM parallel 

to V. From draw lines OD, 

ON, OM to meet CM. Draw 

CB parallel to OP. From tha 

points 1, 5, 6, where OD, ON, 

OM intersect CB, draw lines 

1 2, 5 7, 6 8 parallel to OV. From D, N, M, draw lines perpendicular to OV. 

The points of intersection 2, 7, 8 are points on the required curve. Proof : draw 
EC, jF6, parallel to OV, and 8 A parallel to OP. In the similar tri- 
angles OQB, OMA, we have 6 B: MA : : OB : OA, or 8 A : CB : : EC : FS, 
whence 8 A x F8 = CB x EC, or P 8 V 8 = P C V C . 




Fig. 24. 



Art. 92, 95. — Construction of Equilateral 
Hyperbola. 




Fig. 25. 



93. Alternative Method. In Fig. 25 let & be a known point on the 
curve. Draw aD through b and lay off DA = ab. Then A is another 
point on the curve. Additional points may be found by either of the 
constructions indicated: e.g. by drawing dh and laying off Tif=db, 
or by drawing BK and laying off Kf= BA. These methods are prac- 
tically applied in the examination of the expansion lines of steam 
engine indicator diagrams. 



94. Theorem : Along an isothermal path for a per- 
fect gas, the external work done is equivalent to 
the heat absorbed (Art. 78). 

v The internal energy 
is unchanged, as indi- 
cated by Joule's law 



J K 



I a 



Art. 93. — Second Method for Plotting 
Hyperbolas. 



(Art. 75) ; hence the expenditure of heat is solely for the performance 

of external work. H= T+ 1+ W, but T = 0, T+I=0, and H= W. 

Conversely, we have Mayer's principle, that " the work done in compressing a 
portion of gas at constant temperature from one volume to another is dynamically 
equivalent to the heat emitted by the gas during the compression " (2). 

95. Work done during Isothermal Expansion. To obtain the ex- 
ternal work done under any portion of the isothermal curve, Fig. 24, 
we must use the integral form, 

W= ( v PdV 



46 APPLIED THERMODYNAMICS 

in which v, I^are the initial and final volumes. But, from the equa- 
tion of the curve, pv = PV, P = pv -r- V, and when p and vare given, 



w 



dV , F ^ F 



pv^-y^pvioge— =Htlog e — =Btlog e £- 



The heat absorbed is equal to this value divided by 778. For V = infinity, this 
expression is itself equal to infinity ; the external work area uuder an indefinitely 
extended isothermal is infinite. 

96. Perfect Gas Isodynamic (Art. 8T). Since in a perfect gas the 
internal energy is fixed by the temperature alone, the internal energy 
along an isothermal is constant, and the isodynamic and isothermal 
paths coincide. 

97. Expansion in General. We may for the present limit the 
consideration of possible paths to those in which increases of volume 
are accompanied by more or less marked decreases in pressure ; the 
latter ranging, say, from zero to infinity in rate. If the volume in- 
creases without any fall in pressure, the 
path is one of constant pressure ; if the 
volume increases only when the fall of 
pressure is infinite, the path is one of con- 
stant volume. The paths under considera- 
tion will usually fall between these two, 

Fig. 26. Art. 07. -Expansive like ao i ac, ad, etc., Fig. 26. The general 
Paths - law for all of these paths is PP = a con- 

stant, in which the slope is determined by the value of the exponent n 
(Art. 91). For n = 0, the path is one of constant pressure, ae, Fig. 26. 
For n— infinity, the path is one of constant volume. The "steepness" 
of the path increases with the value of n. (Note that the exponent 
n applies to V only, not to the whole expression.) 

98. Work done by Expansion. For this general case, the external 
work area, adopting the notation of Art. 95, is, 

W= tj^PdV; 

But since pv n — PV n , P = pv n -=- V n ; whence, when;? and v are given, 
1 Jv l—n\ J Ti — 1 n— 1 



p 


a 




CONSTAf. 


T 


PRESSURE 


n = o c 


in 

s 

=> 

O 

> 










-8 




1- 
z 

O 

a 


8 






^^^-d 


c 












V 



THE ADIABATIC 47 

When 7 = infinity, P = 0, and the work is indeterminate by this expression ; but 
we may write W = -&- (l - £Z\ = -J™_.Fl - f-^)"" 1 ], ^ which, for 7= in- 
finity, W = pv -f- (n — 1), a finite quantity. The work under an exponential curve 
is thus finite and commensurable, no matter how far the expansion be continued. 
For n — 1, the work obviously becomes infinite with infinite expansion (Art. 95). 

99. Relations of Properties. For a perfect gas, in which — — =•££, we have 

PVt = pvT. 

If expansion proceeds according to the law PV n = pv n , we obtain, dividing the 
first of these equations by the second, 

Zj = It, whence! = ( "V"". 
7* u n T \Vl 

This result permits of the computation of the change in temperature following a 
given expansion. We may similarly derive a relation between temperature and 
pressure. Since 

i i 

pv n = PV n , v(p) n = V(P) n . Dividing the expression pvT — PVt by this, we have 



T(p) n =t(P) n , whence ± -- 

By interpretation of these formulas of relation, we observe that for 
values of n exceeding unity, during expansion {i.e. increase of volume), the 
pressure and temperature decrease, while external work is done. The 
gain or loss of heat we cannot yet determine. On the other hand, during 
compression, the volume decreases, the pressure and temperature increase, 
and work is spent upon the gas. In the work expression of Art. 98, if 
p, v, t are always understood to denote the initial conditions, and P, V, T, 
the final conditions, then the work quantity for a compression is negative. 

100. Adiabatic Process. This term (Art. 83) is applied to any 
process conducted without the reception or rejection of heat from or 
to surrounding bodies by the substance under consideration. It is 
by far the most important mode of expansion which we shall have to 
consider. The substance expands without giving heat to, or taking 
heat from, other bodies. It may lose heat, by doing work ; or, in com- 
pression, work may be expended on the substance so as to cause it to 
gain heat : but there is no transfer of heat between it and surrounding 
bodies. If air could be worked in a perfectly non-conducting cylinder, 
we should have a practical instance of adiabatic expansion. In 
practice we sometimes approach the adiabatic path closely, by causing 
expansion to take place with great rapidity, so that there is no time 



48 APPLIED THERMODYNAMICS 

for the transfer of heat. The expansions and compressions of the air 
which occur in sound waves are adiabatic, on account of their rapidity 
(Art. 105). In the fundamental equation H ' — T + I + W, the adi- 
abatic process makes H= 0, whence W= — (T + I) ; or, the external 
work done is equivalent to the loss of internal energy, at the expense of 
which energy the work is performed. 

101. Adiabatic Equation. Let unit quantity of gas expand adiabatically 
to an infinitesimal extent, increasing its volume by dv, and decreasing its 
pressure and temperature by dp and dt. As has just been shown, 
W== — (T + I), the expression in the parenthesis denoting the change in 
internal energy during expansion. The heat necessary to produce this 
change would be Idt, I being the specific heat at constant volume. The ex- 
ternal work done is W= pdv; consequently, pdv = — Idt. From the 

equation of the gas, pv = Rt, t=£-, whence, dt = — (pdv -f vdp). Using 
this value for dt, 

pdv = (pdv -f vdp). 

i R 

But R is equal to the difference of the specific heats, or to k — I ; so that 

pdv — — - (pdv + vdp), 

k — I 

c 
ypdv — pdv = — pdv — vdp, 

y~ — = — -±, giving by integration, 
v p 

ylog e v + log e p = constant, 

or pv y = constant, 

y being the ratio of the specific heats at constant pressure and con- 
stant volume (Art. 69.) 

102. Second Derivation. A simpler, though less satisfactory, mode of 
derivation of the adiabatic equation is adopted by some writers. Assum- 
ing that the adiabatic is a special case of expansion according to the law 
pv n = PV n , the external work done, according to Art. 98, is 

R(t-T) 



ADIABATIC EXPANSION 49 

During a change of temperature from t to T, the change in internal energy- 
is l(t — T), or from Art. 70, since I = R -h (y — T), it is 

R(t - T) 

y-i 

But in adiabatic expansion, the external ivork done is equivalent to the 
change in internal energy ; consequently 

R(t- T) = B(t- T) , 
n— 1 y — 1 

n = y, and the adiabatic equation is pv* = PV y . For air, the adiabatic is 
then represented by the expression p(yf' m = a constant. 

103. Graphical Presentation. Since along an adiabatic the external 
work is done at the expense of the internal energy, the temperature must 
fall during expansion. In the diagram of Fig. 19, this is shown by com- 
paring the line ab, an isothermal, with ae, an adiabatic. The relation of 
p to v, in adiabatic expansion, is such as to cause the temperature to fall. 
The projections of these two paths on the pv plane show that as 
expansion proceeds from a, the pressure falls more rapidly along 
the adiabatic than along the isothermal, a result which might have been 
anticipated from comparison of the equations of the two paths. If an 
isothermal and an adiabatic be drawn through the same point, the latter 
will be the " steeper " of the two curves. Any number of adiabatics may 
be constructed on the pv diagram, depending upon the value assigned to 
the constant (pv y ) ; but since this value is determined, for any particular 
perfect gas, by contemporaneous values of p and v, only one adiabatic can 
be drawn for a given gas through a given point. 

104. Relations of Properties. By the methods of Art. 98 and 
Art. 99, we find, for adiabatic changes, 

During expansion, the pressure and temperature decrease, external work is done 
at the expense of the internal energy, and there is no reception or rejection of heat. 

105. Direct Calculation of the Value of y. The velocity of a wave in an 
elastic medium is, according to a fundamental proposition in dynamics, directly 
as the square root of the coefficient of elasticity divided by the density of the 
medium ; or, for ultimate values, 

v = Ve -^ d. 

Let g denote the acceleration due to gravity, w the weight of unit volume of the 
medium at the density d, m the weight of unit volume of mercury, and b the 



50 



APPLIED THERMODYNAMICS 



height of the mercurial barometer. For unconfined air at constant pressure, 
the pressure equals the elasticity ; for Idp — — eds, in which dp is an infinitesimal 
increment of pressure applied to a body of length /, producing an extension ds, 
equivalent to a compression — ds; and if the body be a gas kept at constant tem- 
perature, pv = c, pdv = - vdp ; and if its form be prismatic and its cross section 
unity, such that I — v, then dv = ds, pds = — Idp, and p = — Idp -f- ds = e. Then 
e =p = bin, and since d — w ~ g, we have 

v = Vbmg ~ w. 

This would be the velocity of sound in air, for example, if there were no change 
in temperature. But the vibrations which constitute sound are accompanied by 
changes in temperature; these changes are adiabatic (Art. 100), and it has been 
shown (Art. 104), that the pressure varies during such changes inversely as that 
power of the specific volume whose exponent is y; or directly as the y power of 
the density. Then e=(f)d y . Taking the expression first given, and putting in 
differential form, 



de_ 
^dd 



But if e = (f)d y , we have, say, e— ad y , de = yad y ~ x dd, a 



e -T- d y , de 



y- • dd 



= -^.^,and 

10 



Ibmyg 
w 



At the temperature of melting ice, when b = 2.494 ft., v has been found by experi- 
ment to be 1089 ft. per second ; whence 

0.081 x 1089 x 1089 



y = 



bmg 32.19 x 2.494 x 8-19.3 



1.411. 



(3) 



106. Representation of Heat Absorbed. Theorem : The heat ab- 
sorbed on any path is represented on the PV diagram by the area en- 
closed between that path and the two adiabatics through its extremities, 
indefinitely prolonged to the right. 

Let the path be ab, Fig. 27. Draw the adiabatics an, IN. These 
may be conceived to meet at an infinite dis- 
tance to the right, forming with the path the 
closed cycle abNn. In such closed cycle, 
the total expenditure of heat is, from Art. 
90, represented by the enclosed area ; but 
since no heat is absorbed or emitted along 
Fig. 27. Arts, inn, ion.— Rep- the adiabatics, all of the heat changes in the 

reservation of Heat Ab- C y C i e mus t have occurred along" the path ah* 
sorbed. J ox 

and this change of heat is represented by the 
area abNn. If the path be taken in the reverse direction, i.e. from b 
to a, the area abNn measures the heat emitted. 




GRAPHICAL REPRESENTATIONS 



51 




107. Representations of Thermal Capacities. Let ab, cd, Fig. 28, be two 
isothermals, differing by one degree. Then efnN represents the specific 
heat at constant volume, egmN the specific heat at 

constant pressure , eN, fn, and gm being adiabatics. 
The latter is apparently the greater, as it should 
be. Similarly, if ab denotes unit increase of 
volume, the area abMN represents the latent heat 
of expansion. The other thermal capacities men- 
tioned in Art. 58 may be similarly represented. 

Fig. 28. Art. 107. — Thermal 
Capacities. 

108. Isodiabatics. Let AD, Fig, 29, represent 

any path following the law pv n = PV n , intersecting the two isothermals 

CX, B Y. The heat absorbed along this 
path may be represented by the area 
nADN, An and DN being adiabatics. If 
some other path, BC, be found, in which 
pv' 1 = PV U , the value of n being the 
same as that for the path AD, this path 
connecting the same two isothermals, 
then the two paths AD and BC are 
called isodiabatics, and as will appear 
(Art. 112), the areas mCBM and nADN 
are equal. 

Theorem. The ratios of pressures or of volumes at points on isodiabatics 
intersected by isothermals are constant. 

In Fig. 29, if we designate the pressures at A, D, C, and B by P A , P D , 
P c , P B , respectively, then from Art. 99, 




Fig. 29. Art. 108. — Isodiabatics. 






So also, 



TV 1 " 



whence — - 

V A 



la 

v B 



109. Derivation of Joule's Law. From the theorem of Art. 106, Rankine 
has established in a very simple manner the principle of Joule, that the 
change of internal energy along any path of any substance depends upon the 
initial and final states alone, and not upon the nature of the path. In 
Fig. 27, draw the vertical lines ax, by. The total heat absorbed along 
ab = nabN, the external work done = xaby. The difference = nabN — xaby 
= nzbN— xazy, is the change in internal energy ; H= T + 1+ W, whence 
H— W = (T-\- 1) ; and the extent of these areas is unaffected by any 
change in the path ab, so long as the points a and b remain fixed. 




52 APPLIED THERMODYNAMICS 

This demonstration is of major importance because it establishes 
the cardinal nature of internal energy for all substances in uniform 
thermal condition. Compare Art. 90, footnote. 

110. Value of y. A method of computing the value of y for air has 
been given in Art. 105. The apparatus shown in Fig. 30 has been used 
by several observers to obtain direct values for various gases. The vessel 
was filled with gas at P, V, and T, T being the temperature of the atmos- 
phere, and P a pressure somewhat in excess of that 
of the atmosphere. By opening the stopcock, a 
sudden expansion took place, the pressure falling 
to that of the atmosphere, and the temperature 
falling to a point considerably below that of the 
atmosphere. Let the state of the gas after this 
adiabatic expansion be p, v, t. Then, since 

Fig. 30. Art. 110. -De- =P ^ y _ logj>-logP , 

sormes' Apparatus. log V — log V 

After this operation, the stopcock is closed, and the gas remaining in the 
vessel is allowed to return to its initial condition of temperature, T. 
During this operation, the volume remains constant; so that the final 
state isi> 2 , v, T; whence p 2 v — PV, or log V — log v = logp 2 — log P. Sub- 
stituting this value of log V — log v in the expression for y, we have 

y== logp~logP ^ 
logp 2 -logP' 

so that the value of y may be computed from the pressure changes alone. 
Clement and Desormes obtained in this manner for air, y = 1.3524 ; G-ay- 
Lussac and Wilter found ?/ = 1.3745. The experiments of Hirn, Weisbach, 
Masson, Cazin, and Kohlrausch were conducted in the same manner. The 
method is not sufficiently exact. 

11L Expansions in General. In adiabatic expansion, the external work 
done and the change in internal energy are equally represented by the 

a™. py 

expression 2 - , derived as in Art. 98. For expansion from p, v to 

y — 1 

u pv 

infinite volume, this becomes _^ • The external work done during any 

PV 



expansion according to the law pv n = PV n from pv to PV, is W=±— 



n — 1 

The stock of internal energy at p, v, is — - — = It ; at P, V, it is — IT. 

2/-1 y- 1 

The total heat expended during expansion is equal to the algebraic sum 
of the external work done and the internal energy gained. Then, 



POLYTROPIC PATHS 53 

= ^,_ 1)(J _^_L i __L_) =Kt _ r) (ti_i 

= l(t— T)( ^~ n \ in which t is the initial, and T the final temperature. 
\n - ly 

This gives a measure of the net heat absorbed or emitted during any ex- 
pansion or compression according to the law pv n = constant. When n 
exceeds y, the sign of H is minus ; heat is emitted ; when n is less than y 
but greater than 1.0, heat is absorbed : the temperature falling in both 
cases. When n = y, the path is adiabatic, and heat is neither absorbed 
nor emitted. 

112. Specific Heat. Since for any change of temperature involving 
a heat absorption H, the mean specific heat is 

H 
T-t' 

we derive from the last equation of Art. Ill the expression, 

. = »*=£ 

n — 1 

giving the specific heat along any path pv n = PV n . Since the values 
of n are the same for isodiabatics, the specific heats along such paths are 
equal (Art. 108). 

113. Ratio of Internal Energy Change to External Work. For any given 
value of n, this ratio has the constant value 



2T-1 

114. Polytropic Paths. A name is needed for that class of paths 
following the general law pv n = P V n , a constant. Since for any 
gas y and I are constant, and since for any particular one of these 
paths n is constant, the final formula of Art. Ill reduces to 

5"= (/)(<- 7). 

In other words, the rate of heat absorption or emission is directly pro- 
portional to the temperature change ; the specific heat is constant. Such 
paths are called polytropic. A large proportion of the paths exempli- 
fied in engineering problems may be treated as polytropics. 



54 



APPLIED THERMODYNAMICS 



115. Relations of n and s. We have discussed such paths in which the 
value of n ranges from 1.0 to infinity. Figure 31 will make the concep- 
tion more general. Let a represent the initial condition of the gas. If 




Fig. 31. Art. 115.— Polytropic Paths. 

it expands along the isothermal ab, n = 1, and s, the specific heat, is infi- 
nite ; no addition of heat whatever can change the temperature. If it 
expands at constant pressure, along ae, n = 0, and the specific heat is finite 
and equal to ly = k. If the path is ag, at constant volume, n is infinite 
and the specific heat is positive, finite, and equal to I. Along the isother- 
mal af (compression), the value of n is 1, and s is again infinite. Along 
the adiabatic ah, n = 1.402 and s = 0. Along ai, n = and s = k. Along 
ad, n is infinite and s = I. Most of these relations are directly derived 
from Art. 112, or may in some cases be even more readily apprehended by 
drawing the adiabatics, en, gN, fm, iM, dp, bP, and noting the signs of the 
areas representing heats absorbed or emitted with changes in temperature. 
For any path lying between ah and af or between ac and ab, the specific 
heat is negative, i.e. the addition of heat cannot keep the temperature from fall- 
ing : nor its abstraction from rising. 



116. Relations of Curves : Graphical Representation of n. Any number of 
curves may be drawn, following the law pv n = C, as the value of C is changed. 



RELATIONS OF n AND s 



55 



In Fig. 32, let ab, cd, efbe curves thus drawn. Their general equation is pi 



whence 



npdv 



+ dp = 



or 



_dp 

dv v 

If MTV is the angle made by 
the tangent to one of the curves 
with the axis OV, and MOV 
the angle formed by the radius 
vector RM with the axis OV, 
then, since dp -*- dv is the tan- 
gent of MTV, and j9 -f- v is the 
tangent of MOV 




Fig. 32. Art. 116.- — Determination of Exponent. 
tan MTV =n tan MOV. 



If the radius vector be produced as RMNQ, the relations of the angles made be- 
tween the OF axis and the successive tangents MT, N-S, QU, are to the angle 
p MOV as just given; hence the various tangents 

are parallel (4). 

Since tan MTV = Mg -=- gT and tan MOV ' = 
Mg + Og, the preceding equation gives 

_ Mg _ n Mg 




9T 



Og 



Art. 116. — Negative 
Exponent. 



radius vector, then by similar triangles 
OgygT.'.OB : OA and Og - gT = OB = n. 
Figure 33 illustrates the generality of this 
method by showing its application to a 
curve in which the value of n is negative. 

117. Plotting of Curves: Brauer's 
Method. The following is a simple method 
for the plotting of exponential curves, in- 
cluding the adiabatic, which is ordinarily 
a tedious process. Let the point M, 
Fig, 34, be given as one point on the re- 
quired curve. Draw a line OA making an 
angle VOA with the axis OV, and a line 
OB making an angle FOB with the axis 



whence n = Og -=- gT. (The algebraic signs of 
Og and gT, measured from g, are different.) In 
order to determine the value of n from a given 
curve, we need therefore only draw a tangent 
MT and a radius vector MO, whence by drop- 
ping the perpendicular Mg the relation Og + gT 
is established. If we lay off from the distance 
OA as a unit of length, drawing A C parallel to 
the tangent, and CB through C, parallel to the 
p 




Fig. 34. Art. 117. — Brauer's Method. 



56 



APPLIED THERMODYNAMICS 



OP. Draw the vertical line MS and the horizontal line MT. Also draw the 
line TU making an angle of 45° with OP, and the line SR making an angle of 
45° with MS. Draw the vertical line RN through R, and the horizontal line UN 
through U. The coordinates of the point of intersection, N, of these lines, are 
OR and RN. Let the coordinates of M, TM ( = OQ), and MQ be designated by 
v, p ; and those of N, OR, and RN ( = 0L), by V, P. Then tan VOA = QS - OQ 
= QR + TM = (V-v) + v, and tan POB = UL - 0L= TL + NR = (p - P) -P; 
whence V = v (tan VOA + 1) and /? = P (tan P0£ + 1). If the law of the 
curve through M and N is to be pv n = P V n , we obtain 

P (tan POB + l)v n = P{y(tan VOA + l)} n , 

whence (tan POB + 1) = (tan VOA + l) w . If now, in the first place, we make the 
angles POB, VOA such as to fulfill this condition, then the point N and others 
similarly determined will be points on a curve following the. law joy* = PV n . 

118. Tabular Method. The equation pv n = PV n may be written p = P[—j 

or log jo — log P = n log ( V ■+■ v). If we express P as a definite initial pressure for 
all PV n curves, then for a specific value of n and for definite ratios V -i- v we may 
tabulate successive values of logjo and of p. Such tables for various values of n 
are commonly used. In employing them, the final pressure is found in terms of 
the initial pressure for various ratios of final to initial volume. 

119. Representation of Internal Energy. In- Fig. 35, let An represent 
an adiabatic. During expansion from A to a, the external work done is 

Aabc, which, from the law of the adiabatic, is 
equal to the expenditure of internal energy. If 
expansion is continued indefinitely, the adiabatic 
An gradually approaches the axis OV, the area 
below it continually representing expenditure of 
internal energy, until with infinite expansion An 
and OF coincide. The internal energy is then ex- 
hausted. The total internal energy of a substance 
may therefore be represented by the area between 
the adiabatic through its state, indefinitely prolonged 

to the right, and the horizontal axis. Eepresenting this quantity by E, then 

from Art. Ill, 




Fig. 35. Art. 119.— Repre- 
sentation of Internal 
Energy. 



E 






pv 



y 



where v is the initial volume, p the initial pressure, and y the adiabatic 
exponent. This is a finite and commensurable quantity. 



120. Representation by Isodynamic Lines. A defect of the preceding 
representation is that the areas cannot be included on a finite diagram. 



GRAPHICAL REPRESENTATIONS 



57 



In Fig. 36, consider the path AB. Let BC be an adiabatic and AC an- 
isodynamic. It is required to find the change of internal energy between 
A and B. The external work done during adi- 
abatic expansion from B to C is equal to BCcb ', 
and this is equal to the change of internal en- 
ergy between B and C. But the internal energy 
is the same at C as at A, because AC is an 
isodynamic. Consequently, the change of in- 
ternal energy between A and B is represented 
by the area BCcb ; or, generally, by the area 
included between the adiabatic through the final 
state, extended to its intersection with the iso- 
dynamic through the initial state, and the hori- 
zontal axis. 




Fig. 36. Arts. 120, 121. — In- 
ternal Energy, Second Dia- 
gram. 



121. Source of External Work. If in Fig. 36 the path is such as to increase 
the temperature of the substance, or ev r en to keep its 
temperature from decreasing as much as it would 
along an adiabatic, then heat must be absorbed. 
Thus, comparing the paths ad and ac, Fig. 37, aN 
and cm being adiabatics, the external work done 
along ad is adef, no heat is absorbed, and the internal 
energy decreases by adef. Along ac, the external 
work done is acef, of which ar/e/was done at the ex- 
pense of the internal energy, and acd by reason of 
the heat absorbed. The total heat absorbed was 

Nacm, of which acd was expended in doing external work, while Ndcm went 

to increase the stock of internal energy. 




Fig. 37. Art. 121. —External 
Work and Internal Energy. 



122. Application to Isothermal Expansion. If the path is isothermal, Fig. 38 
line AB, then if BN, An are adiabatics, we have, 

W + X = external work done, 

X + Y = heat absorbed = W + X, 

W + Z = internal energy at A, 

Y + Z = internal energy at B, 

W = work done at the expense of the in- 
ternal energy present at A, 

X = work done by reason of the absorption 
of heat along AB, 

Z = residual internal energy of that originally 
present at A, 

Y— additional internal energy imparted by 
the heat absorbed; 
and since in a perfect gas isothermals are isodynamics, we note that 

W + Z = Y+ Z and W = F (5). 




Fig. 38. Art. 122. —Heat and 
Work in Isothermal Expansion. 



58 



APPLIED THERMODYNAMICS 




123. Finite Area representing Heat Expenditure. In Fig. 39, let ab be any 

path, bn and aN adiabatics, and ac an isodynamic. The external work done along 

ab is abde ; while the increase of internal energy is 
bcfd. The total heat absorbed is then represented by 
the combined areas abcfe. If the path ab is iso- 
thermal, this construction leads to the known result 
that there is no gain of internal energy, and that the 
total heat absorbed equals the external work. If the 
path be one of those de- 
scribed in Art. 115 as of 
negative specific heat, we 
may represent it as ag, 
Fig. 40. Let bgm be an 
adiabatic. The external 
work done is agde. The change of internal energy, 
from Art. 120, is bgdf, if ab is an isodynamic; and 
this being a negative area, we note that internal en- 
ergy has been expended, although heat has been ab- 
sorbed. Consequently, the temperature has fallen. It 
seems absurd to conceive of a substance as receiving heat while falling in tem- 
perature. The explanation is that it is cooling, by doing external work, faster 
than the supply of heat can warm it. Thus, H = T + I + W; but H< W; con- 
sequently, ( T + I) is negative. 



Fig. 39. Art. 123.— Represeu 
tation of Heat Absorbed. 




Fig. 40. Art. 123. — Nega- 
tive Specific Heat. 



Modifications in Irreversible Processes 

124. Constrained and Free Expansion. In Art. 86 it was assumed that 
the path of the substance was one involving changes of volume against a 
resistance. Such changes constitute constrained expansion. In this pre- 
liminary analysis, they are assumed to take place slowly, so that no 
mechanical work is done by reason of the velocity with which they are 
effected. When a substance expands against no resistance, as in Joule's 
experiment, or against a comparatively slight resistance, w r e have what is 
known as free expansion, and the external work is wholly or partly due 
to velocity changes. 

125. Reversibility. All of the polytropic curves which have thus far 
been discussed exemplify constrained expansion. The external and in- 
ternal pressures at any state, as in Art. 86, differ to an infinitesimal 
extent only ; the quantities are therefore in finite terms equal, and the 
processes may be worked at ivill in either direction. A polytropic path 
having a finite exponent is in general, then, reversible, a characteristic of 
fundamental importance. During the adiabatic process which occurred 
in Joule's experiment, the externally resisting pressure was zero while 
the internal pressure of the gas was finite. The process could not be 




IRREVERSIBLE PROCESSES 59 

reversed for it would be impossible for the gas to flow against a pressure 
greater than its own. The generation of heat by friction, the absorption 
of heat by one body from another, etc., are more familiar instances of 
irreversible process. Since these actions take place to a greater or less 
extent in all actual thermal phenomena, it is impossible for any actual 
process to be perfectly reversible. "A process affecting two substances is 
reversible only when the conditions existing at the commencement of the 
process may be directly restored without compensating changes in other 
bodies." 

126. Irreversible Expansion. In Fig. 41, let the substance expand 
unconstramedly, as in Joule's experiment, from a to b, this expansion 
being produced by the sudden decrease in ex- 
ternal pressure when the stopcock is opened. 
Along the path ab, there is a violent movement of 
the particles of gas; the kinetic energy thus 
evolved is transformed into pressure at the end 
of the expansion, causing a rise of pressure to c. 
The gain or loss of internal energy depends solely 

upon the states a, c; the external work done does Fig. 41. Art. 126.-Irr 
not depend on the irreversible path ab, for with versible Path, 

a zero resisting pressure no external work is done. The theorem of Art. 86 
is true only for reversible operations. 

127. Irreversible Adiabatic Process. Careful consideration should be 
given to unconstrained adiabatic processes like those exemplified in Joule's 
experiment. In that instance, the temperature of the gas was kept up by 
the transformation back to heat of the velocity energy of the rapidly 
moving particles, through the medium of friction. We have here a special 
case of heat absorption. No heat was received from without; the gas 
remained m a heat-insulated condition. While the process conforms to 
the adiabatic definition (Art. 83), it involves an action not contemplated 
when that definition was framed, viz, a reception of heat, not from sur- 
rounding bodies, but from the mechanical action of the substance itself. The 
fundamental formula of Art. 12 thus becomes 

H=T + I+ W+V, 
in which Fmay denote a mechanical effect due to the velocity of the parti- 
cles of the substance. This subject will be encountered later in important 
applications (Arts. 175, 176, 426, 513). 

1893 (1 n ZT 0d ^Z iCS ' T' P " ^ (2) Alexander > T reatise on Thermodynamics, 

03 RaJl ™ o i thermodynamics, 123; Alexander, Thermodynamics, 

103 Rankine, The Steam Engine, 249, 321; Wood, Thermodynamics. 71-77, 437 

K^ K]ein tr " r ' 156 ' «> "*-. *- w 



60 APPLIED THERMODYNAMICS 



SYNOPSIS OF CHAPTER V 

Pressure, volume and temperature as thermodynamic coordinates. 

Thermal line, the locus of a series of successive states ; path, a projected thermal line. 

Paths: isothermal, constant temperature; isodynamic, constant internal energy ; 

adiabatic, no transfer of heat to or from surrounding bodies. 
The geometrical representation of the characteristic equation is a surface. 
The P V diagram : subtended areas represent external work ; a cycle is an enclosed 

figure ; its area represents external work ; it represents also the net expenditure of 

heat. 
The isothermal : pv n = c, in which n = 1, an equilateral hyperbola ; the external work 

V 

done is equivalent to the heat absorbed, = pv \og e — : with a perfect gas, it coin- 

v 
cides with the isodynamic. 

mjj py t /y\l—H J 7 / p M-l 

Paths in general : pv' 1 = c : external work = 2 - ;_=[ — ] ; _ — [ _ \ « . 

y * n-l T \V) t \p) 

The adiabatic : the external work done is equivalent to the expenditure of internal 
energy ; pvv — c ; y — 1 .402 ; computation from the velocity of sound in air. 

The heat absorbed along any path is represented by the area between that path and 
the two projected adiabatics ; representation of k and I. 

Isodiabatics : n\=n%\ equal specific heats. 

Rankine's derivation of Joule's law : the change of internal energy between two states 
is independent of the path. 

Apparatus for determining the value of y from pressure changes alone. 

Along any path pv n = c, the heat absorbed is l{t — T)l y ~^ j . the mean specific heat 

is I T-- Such paths are called polytropics. Values of n and s for various paths. 

n — 1 

Graphical method for determining the value of n; Brauer's method for plotting poly- 
tropics ; the tabular method. 

Graphical representations of internal energy ; representations of the sources of external 
work and of the effects of heat ; finite area representing heat expenditure. 

Irreversible processes : constrained and free expansion ; reversibility ; no actual pro- 
cess is reversible ; example of irreversible process ; subtended areas do not repre- 
sent external work ; in adiabatic action, heat may be received from the mechanical 
behavior of the substance itself; H= T + 1 + W -\- V. 



PROBLEMS 

1. On a perfect gas diagram, the coordinates of which are internal energy and 
volume, construct an isodynamic, an isothermal, and an isometric path through E = 2, 
V=2. 

2. Plot accurately the following: on the TV diagram, an adiabatic through 
T=270, F=10; an isothermal through T=S00, F=20: on the TP diagram, an 
adiabatic through T=230, P - 5 ; an isothermal through T=190, P=30. On the 
EV diagram, show the shape of an adiabatic through E — 240, V — 10. 

3. Show the isometric path of a perfect gas on the PT plane j the isopiestic, on 
the VT plane. 



PROBLEMS 61 

4. Sketch the TFpath of wax from 0° to 290° F., assuming the melting point to 
be 90°, the boiling point 290°, that wax expands in melting, and that its maximum 
density as liquid is at the melting point. 

5. A cycle is bounded by two isopiestic paths through P = 110, P = 100 (pounds 
per square foot), and by two isometric paths through F=20, V = 10 (cubic feet). 
Find the heat expended by the working substance. 

6. Air expands isothermally at 32° F. from atmospheric pressure to a pressure of 
5 lb. absolute* per square inch. Find its specific volume after expansion. 

7. Given an isothermal curve and the OF axis, find graphically the OP axis. 

8. Prove the correctness of the construction described in Art. 93. 

9. Find the heat absorbed during the expansion described in Problem 6. 

10. Find the specific heat for the path PV 12 = c, for air and for hydrogen. 

11. Along the path PV 12 = c, find the external work done in expanding from 
P= 1000, V— 10, to V ' = 100. Find also the heat absorbed, and the loss of internal 
energy, if the substance is one pound of air. Units are pounds per square foot and 
cubic feet. 

12. A perfect gas is expanded from p = 400, v = 2,t = 1200, to P = 60, V= 220. 
Find the final temperature. 

13. Along the path PV 12 = c, a gas is expanded to ten times its initial volume of 
10 cubic feet per pound. The initial pressure being 1000, and the value of B 53.36, 
find the final pressure and temperature. 

14. Through what range of temperature will air be heated if compressed to 10 at- 
mospheres from normal atmospheric pressure and 70° F., following the \&wpv 1B = c ? 
What will be the rise in temperature if the law is pv y = c ? If it is pv = c ? 

15. Find the heat imparted to one pound of this air in compressing it as described 
according to the law^w 1 - 3 = c, and the change of internal energy. 

16. In Problem 14, after compression along the path pv 1 - 3 = c, the air is cooled 
at constant volume to 70° F., and then expanded along the isodiabatic path to its initial 
volume. Find the pressure and temperature at the end of this expansion. 

17. The isodiabatics ab, cd are intersected by lines of constant volume ac, bd. 
Prove — « = ^ and -^ = Th. 

Pc Pd T e Ta 

18. In a room at normal atmospheric pressure and constant temperature, a 
cylinder contains air at a pressure of 1200 lb. per square inch. The stopcock on the 
cylinder is suddenly opened. After the pressure in the cylinder has fallen to that of 
the atmosphere, the cock is closed, and the cylinder left undisturbed for 24 hours. 
Compute the pressure in the cylinder at the end of this time. 

19. Find graphically the value of n for the polytropic curve ab, Fig. 41. 

20. Plot by Brauer's method a curve pv 1 - 8 = 26,200. Use a scale of 1 inch per 
4 units of volume and per 80 units of pressure. Begin the curve with p = 1000. 

* Absolute pressures are pressures measured above a perfect vacuum. The absolute 
pressure of one standard atmosphere is 14.697 lb. per square inch. 



62 APPLIED THERMODYNAMICS 

21. Supply the necessary figures in the following blank spaces, for n — 1.8, and 
apply the results to check the curve obtained in Problem 20. Begin with v = 6.12, 
p = 1000. 

— = 2.0, 2.25, 2.50, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0 
v 

log £-—n log — = 
P v 

JL = 
P 
P = 

22. The velocity of sound in air being taken at 1140 ft. per second at 70° F. and 
normal atmospheric pressure, compute the value of y for air. 

23. Compute the latent heat of expansion (Art. 58) of air at atmospheric pressure 
and 32° F. 

24. Find the amount of heat converted into work in a cycle 1234, in which 
P 1 = P 4 = 100, Vi = 5, y 4 = 1, P 3 = 30, and the equations of the paths are as follows : 
for 41, PV° = c : for 12, PV y = c ; for 32, PV= c ; for 43, PF 1 - 6 = c. The working 
substance is one pound of air. Find the temperatures at the points 1, 2, 3, 4. 

25. Find the exponent of the polytropic path, for air, along which the specific heat 
is — k. Also that along which it is — I. Kepresent these paths, and the amounts of heat 
absorbed, graphically, comparing with those along which the specific heats are Jc and Z, 
and show how the diagram illustrates the meaning of negative specific heat. 



CHAPTER VI 



THE CARNOT CYCLE 



128. Heat Engines. In a heat engine, work is obtained from 
heat energy through the medium of a gas or vapor. Of the total 
heat received by such fluid, a portion is lost by conduction from the 
walls of containing vessels, a portion is discharged to the atmosphere 
after the required work has been done, and a third portion disap- 
pears, having been converted into external mechanical work. By 
the first law of thermodynamics, this third portion is equivalent to 
the work done ; it is the only heat actually used. The efficiency of a 
heat engine is the ratio of the net heat utilized to the total quantity of 
heat supplied to the engine, or, of external work done to gross heat 

absorbed; to — = — ^— , in which h denotes the quantity of heat 
rejected by the engine, if radiation effects be ignored. 

129. Cyclic Action. In every heat engine, the working fluid passes 
through a series of successive states of pressure, volume, and temperature ; 
and, in order that operation may be continuous, it is necessary either that 
the fluid work in a closed cycle which may be repeated indefinitely, or 
that a fresh supply of fluid be admitted to the engine to compensate for 
such quantity as is periodically 
discharged. It is convenient to 
regard the latter more usual ar- 
rangement as equivalent to the 
former, and in the first instance 
to study the action of a constant 
body of fluid, conceived to work 
continuously in a closed cycle. 



130. Forms of Cycle. The sev- 
eral paths described in Art. 83, and 
others less commonly considered, sug- 
gest various possible forms of cycle, 
some of which are illustrated in Fig. 
42. Many of these have been given names (1). The isodiabatic cycle, bounded by 
two isothermals and any two isodiabatics (Art. 108), may also be mentioned. 

63 




% h 




Fig. 42. Art. 130, Problem 2. — Possible Cycles. 



64 APPLIED THERMODYNAMICS 

131. Development of the Carnot Cycle. Carnot, in 1824, by describing and 
analyzing the action of the perfect elementary heat engine, effected one of the 
most important achievements of modern physical science (2). Carnot, it is true, 
worked with insufficient data. Being ignorant of the first law of thermodynamics, 
and holding to the caloric theory, he asserted that no heat was lost during the 
cyclic process ; but, though to this extent founded on error, his main conclusions 
were correct. Before his death, in 1832, Carnot was led to a more just conception 
of the true nature of heat ; while, left as it was, his work has been the starting 
point for nearly all subsequent investigations. The Carnot engine is the limit 
and standard for all heat engines. 

Clapeyron placed the arguments of Carnot in analytical and graphical form ; 
Clausius expressed them in terms of the mechanical theory of heat : James Thomp- 
son, Rankine, and Clerk Maxwell corrected Carnot's assumptions, redescribed the 
cyclic process, and redetermined the results ; and Kelvin (3) expressed them in 
their final and satisfactory modern form. 

132. Operation of Carnot's Cycle. Adopting Kelvin's method, 
the operation on the Carnot engine may be described by reference 
to Fig. 43. A working piston moves in the cylinder c, the walls of 

which are non-conduct- 
ing, while the head is 
a perfect conductor. 
The piston itself is 

Fig. 43. Arts. 132, 138.— Operation of the Carnot Cycle. , . , 

a non-conductor and 
moves without friction. The body s is an infinite source of heat 
(the furnace, in an actual power plant) maintained constantly at 
the temperature T, no matter how much heat is abstracted from it. 
At r is an infinite condenser, capable of receiving any quantity of 
heat whatever without undergoing any elevation of temperature 
above its initial temperature t. The plate /is assumed to be a per- 
fect non-conductor. The fluid in the cylinder is assumed to be 
initially at the temperature T of the source. 

The cylinder is placed on s. Heat is received, but the tempera- 
ture does not change, since both cylinder and source are at the 
same temperature. External work is done, as a result of the recep- 
tion of heat ; the piston rises. When this operation has continued 
for some time, the cylinder is instantaneously transferred to the non- 
conducting plate /. The piston is now allowed to rise from the expan- 
sion produced by a decrease of the internal energy of the fluid. It 
continues to rise until the temperature of the fluid has fallen to t, 




THE CARNOT CYCLE 65 

that of the condenser, when the cylinder is instantaneously trans- 
ferred to r. Heat is now given up by the fluid to the condenser, and 
the piston falls ; but no change of temperature takes place. When this 
action is completed (the point for completion will be determined 
later), the cylinder is again placed on /, and the piston allowed to 
fall further, increasing the internal energy and temperature of the 
gas by compressing it. This compression is continued until the 
temperature of the fluid is T and the piston is again in its initial 
position, when the cylinder is once more placed upon s and the opera- 
tion may be repeated. No actual engine could be built or operated 
under these assumed conditions. p 

133. Graphical Representation. The 

first operation described in the preceding 

is expansion at constant temperature. The 

path of the fluid is then an isothermal. 

The second operation is expansion without 

transfer of heat, external work being done 

at the expense of the internal energy ; 

the path is consequently adiabatic. Dur- Fig. 44. Arts. 133-136, 138, 142. 

ing the third operation, we have isothermal e arnot yc e ' 

compression; and during the fourth, adiabatic compression. The 

Carnot cycle may then be represented by abed, Fig. 44. 

134. Termination of Third Operation. In order that the adiabatic compression 
da may bring the fluid back to its initial conditions of pressure, volume, and tem- 
perature, the isothermal compression cd must be terminated at a suitable point d. 
From Art. 99, 

T I V„ 




for the adiabatic da, 

t VTV 

and — = { — 6 for the adiabatic be ; 

t \V C ) 

hence X± = Y± and Ll = Ya- 

V d V e V h V c ' 

that is, the ratio of volumes during isothermal expansion in the first stage must be 
equal to the ratio of volumes during isothermal compression in the third stage, if the 
final adiabatic compression is to complete the cycle. (Compare Art. 108.) 

135. Efficiency of Carnot Cycle. The only transfers of heat dur- 
ing this cycle occur along ab and cd. The heat absorbed along ab is 



66 APPLIED THERMODYNAMICS 

V V 

PaV a \og e —y=RT\og e —^' Similarly, along cd, the heat rejected 

is Rt log e — f • The net amount of heat transformed into work is the 

Yd 

difference of these two quantities ; whence the efficiency, defined in 
Art. 128 as the ratio of the net amount of heat utilized to the total 
amount of heat absorbed, is 

jt = ,- , since ~t _. * f rom Art. 134. 

RTlog e -^ 1 Va ld 

* a 

136. Second Derivation. The external work done under the two adiabatics 
be, da is 

P 6 F 6 -P,F, _„, P a V a 



and 



y-i y -i 

Deducting the negative work from the positive, the net adiabatic work is 

PlV h -P e Vc-PaVa+P*V* . 

y-i 

but P«F a = PbVi, from the law of the isothermal ah; similarly, P d V d = P C V C , and 
consequently this network is equal to zero; and if we express efficiency by the 
ratio of work done to gross heat absorbed, we need consider only the work areas 
under the isothermal curves ab and cd, which are given by the numerator in the 
expression of Art. 135. 

The efficiency of the Carnot engine is therefore expressed by the 
ratio of the difference of the temperatures of source and condenser to 
the absolute temperature of the source. 

137. Carnot's Conclusion. The computations described apply to any sub- 
stance in uniform thermal condition ; hence the conclusion, now universally 
accepted, that the motive power of heat is independent of the agents employed to 
develop it ; it is determined solely by the temperatures of the bodies between which 
the cyclic transfers of heat occur. 

138. Reversal of Cycle. The paths which constitute the Carnot cycle, 
Fig. 44, are polytropic and reversible (Art. 125); the cycle itself is rever- 
sible. Let the cylinder in Fig. 43 be first placed upon r, and the piston 
allowed to rise. Isothermal expansion occurs. The cylinder is trans- 
ferred to /and the piston caused to fall, producing adiabatic compression. 
The cylinder is then placed on s, the piston still falling, resulting in iso- 
thermal compression ; and finally on /, the piston being allowed to rise, so 
as to produce adiabatic expansion. Heat has now been taken from the 



REVERSIBILITY 67 

condenser and rejected to the source. The cycle followed is dcbad, Fig. 44. 
Work has been expended upon the fluid ; the heat delivered to the source s is 
made up of the heat taken from the condenser r, plus the heat equivalent of 
the work done upon the fluid. The apparatus, instead of being a heat 
engine, is now a sort of heat pump, transferring heat from a cold body to 
one warmer than itself, by reason of the expenditure of external work, 
Every operation of the cycle has been reversed. The same quantity of 
heat originally taken from s has now been given up to it ; the quantity 
of heat originally imparted to r is now taken from it ; and the amount of 
external work originally done by the fluid has now been expended upon 
it. The efficiency, based on our present definition, may exceed unity ; it 
is the quotient of heat imparted to the source by work expended. The 
cylinder c must in this case be initially at the temperature t of the con- 
denser r. 

139. Criterion of Reversibility. Of all engines working between the 
same limits of temperature, that which is reversible is the engine of maximum 
efficiency. 

If not, let A be a more efficient engine, and let the power which this 
engine develops be applied to the driving of a heat pump (Art. 138), 
(which is a reversible engine), and let this heat pump be used for restor- 
ing heat to a source s for operating engine A. Assuming that there is no 
friction, then engine A is to perform just a sufficient amount of work to 
drive the heat pump. In generating this power, engine A will consume 
a certain amount of heat from the source, depending upon its efficiency. 
If this efficiency is greater than that of the heat pump, the latter will dis- 
charge more heat them the former receives (see explanation of efficiency, 
Art. 138) ; or will continually restore more heat to the source than engine 
A removes from it. This is a result contrary to all experience. It is 
impossible to conceive of any self-acting machine which shall continually 
produce heat (or any other form of energy) without a corresponding con- 
sumption of energy from some other source. 

140. Hydraulic Analogy. The absurdity may be illustrated, as by Heck (4), 
by imagining a water motor to be used in driving a pump, the pump being em- 
ployed to deliver the water back to the upper level which supplies the motor. 
Obviously, the motor would be doing its best if it consumed no more water than 
the pump returned to the reservoir ; no better performance can be imagined, and 
with actual motors and pumps this performance would never even be equaled. 
Assuming the pump to be equally efficient as a motor or as a pump (i.e. reversible), 
the motor cannot possibly be more efficient. 

141. Clausius' Proof. The validity of this demonstration depends upon the 
correctness of the assumption that perpetual motion is impossible. Since the im- 



68 APPLIED THERMODYNAMICS 

possibility of perpetual motion cannot be directly demonstrated, Clausius estab- 
lished the criterion of reversibility by showing that the existence of a more effi- 
cient engine A involved the continuous transference of heat from a cold body to 
one warmer than itself, without the aid of external agency: an action which is axio- 
matically impossible. 

142. The Perfect Elementary Heat Engine. It follows from the analysis of 
Art. 135 that all engines working in the Carnot cycle are equally efficient ; and 
from Art. 139 that the Carnot engine is one of that class of engines of highest effi- 
ciency. The Carnot cycle is therefore described as that of the perfect elementary 
heat engine. It remains to be shown that among reversible engines working be- 
tween equal temperature limits, that of Carnot is of maximum efficiency. Con- 
sider the Carnot cycle abed, Fig. 44. The external work done is abed, and the 
efficiency, abed ■+- nabN. For any other reversible path than ab, like ae or fb, 
touching the same line of maximum temperature, the work area abed and the heat 
absorption area nabN are reduced by equal amounts. The ratio expressing effi- 
ciency is then reduced by equal amounts in numerator and denominator, and since 
the value of this ratio is always fractional, its value is thus always reduced. For 
any other reversible path than cd, like ch or gd, touching the same line of mini- 
mum temperature, the work area is reduced without any reduction in the gross 
heat area nabN. Consequently the Carnot engine is that of maximum efficiency 
among all conceivable engines worked between the same limits of temperature. A 
practical cycle of equal efficiency will, however, be considered (Art. 257). 

143. Deductions. The efficiency of an actual engine can therefore 
never reach 100 per cent, since this, even with the Carnot engine, would 
require t in Art. 135 to be equal to absolute zero. High efficiency is con- 
ditioned upon a wide range of working temperatures ; and since the mini- 
mum temperature cannot be maintained below that of surrounding bodies, 
high efficiency involves practically the highest possible temperature of 
heat absorption. Actual heat engines do not work in the Carnot cycle ; 
but their efficiency nevertheless depends, though less directly, on the tem- 
perature range. With many working substances, high temperatures are 
necessarily associated with high specific pressures, imposing serious con- 
structive difficulties. The limit of engine efficiency is thus fixed by the 
possibilities of mechanical construction. 

(1) Alexander, Treatise on Thermodynamics, 1893, 38-40. (2) Carnot's Beflec- 
Hons is available in Thurston's translation or in Magie's Second Law of Thermody- 
namics. An estimate of his part in the development of physical science is given by 
Tait, Thermodynamics, 1868, 44. (3) Trans. Boy. Soc. Edinburgh, March, 1851 ; 
Phil. Mag., IV, 1852 ; Math, and Phys. Papers, I, 174. (4) The Steam Engine, I, 
50. 

SYNOPSIS OF CHAPTER VI 

Heat engines : efficiency = heat utilized -f- heat absorbed = — — — = — • 
Cyclic action : closed cycle ; forms of cycle. 



THE CARNOT CYCLE 69 



Carnot cycle : historical development ; cylinder, source, insulating plate, condenser ; 
graphical representation ; termination of third operation, when 



ciency — ^ " • 

Carnot's conclusion : efficiency is independent of the ivorking substance. 

Eeversal of cycle: the reversible engine is that of maximum efficiency; hydraulic 

analogy. 
Carnot cycle not surpassed in efficiency by any reversible or irreversible cycle. 
Limitations of efficiency in actual heat engines. 



PROBLEMS 

1. Show how to express the efficiency of any heat-engine cycle as the quotient 
of two areas on the P V diagram. 

2. Draw and explain six forms of cycle not shown in Fig. 42. 

3. In a Carnot cycle, using air, the initial state is P = 1000, V = 100. The pres- 
sure after isothermal expansion is 500, the temperature of the condenser 200 c F. Find 
the pressure at the termination of the "third operation," the external work done along 
each of the four paths, and the heat absorbed along each of the four paths. Units are 
cubic feet per pound and pounds per square foot. 

4. A non-reversible heat engine takes 1 B. t. u. per minute from a source and is 
used to drive a heat pump having an efficiency (quotient of work by heat imparted to 
source) of 0.70. What would be the rate of increase of heat contents of the source if 
the efficiency of the heat engine were 0.80 ? 

5. Ordinary non-condensing steam engines use steam at 325° F. and discharge it 
to the atmosphere at 215° F. What is their maximum possible efficiency ? 

6. Find the limiting efficiency of a gas engine in which a maximum temperature 
of 3000° F. is attained, the gases being exhausted at 1000° F. 

7. An engine consumes 225 B. t. u. per indicated horse power (33,000 foot-pounds) 
per minute. If its temperature limits are 430° F. and 105° F., how closely does its 
efficiency approach the best possible efficiency ? 



CHAPTER VII 

THE SECOND LAW OF THERMODYNAMICS 

144. Statement of Second Law. The expression for efficiency of 
the Carnot cycle, given in Art. 135, is a statement of the second law 
of thermodynamics. The law is variously expressed ; but, in general, 
it is an axiom from which is established the criterion of reversibility 
(Art. 139). 

With Clausius, the axiom was, 

(a) " Heat cannot of itself pass from a colder to a hotter body; " while the 
equivalent axiom of Kelvin was, 

(b) "It is impossible, by means of inanimate' material agency, to derive 
mechanical effect from any portion of matter by cooling it below the tempera- 
ture of the coldest of surrounding objects." 

With Carnot, the axiom was that perpetual motion is impossible ; while Ran- 
kine's statement of the second law (Art. 151) is an analytical restatement of the 
efficiency of the Carnot cycle. 

145. Comparison of Laws. The law of relation of gaseous properties (Art. 10) 
and the second law of thermodynamics are justified by their results, while theirs* 
law of thermodynamics is an expression of experimental fact. The second law T is a 
" definite and independent statement of an axiom resulting from the choice of one 
of the two propositions of a dilemma" (1). For example, in Carnot's form, we 
must admit either the possibility of perpetual motion or the criterion of reversi- 
bility ; and we choose to admit the latter. The second law is not a proposition to 
be proved, but an " axiom commanding universal assent when its terms are 
understood." 

146. Preferred Statements. The simplest and most satisfactory statement of 
the second law may be derived directly from inspection of the formula for effi- 
ciency, (T — t) -=- T (Art. 135). The most general statement, 

(c) " The availability of heat for doing work depends upon its temperature" leads 
at once to the axiomatic forms of Kelvin and Clausius; while the most specific of 
all the statements directly underlies the presentation of Rankine : 

(d) "If all of the heat be absorbed at one temperature, and 
rejected at another lower temperature, the heat transformed to 

70 



THE SECOND LAW OF THERMODYNAMICS 71 

external work is to the total heat absorbed in the same ratio as that 
of the difference between the temperatures of absorption and rejec- 
tion to the absolute temperature of absorption ;" or, 

H-li = T-t 
H T ' 

in which iZ" represents heat absorbed ; and A, heat rejected. 

147. Other Statements. Forms (a), (6), (c), and (d) are those usually given 
the second law. In modified forms, it has been variously expressed as follows : 

(e) " All reversible engines working between the same uniform tem- 
peratures have the same efficiency." 

(/) " The efficiency of a reversible engine is independent of the nature 
of the working substance." 

(g) " It is impossible, by the unaided action of natural processes, 
to transform any part of the heat of a body into mechanical work, except 
by allowing the heat to pass from that body into another at lower 
temperature." 

(h) "If the engine be such that, when it is worked backward, the 
physical and mechanical agencies in every part of its motions are reversed, 
it produces as much mechanical effect as can be produced by any thermo- 
dynamic engine, with the same source and condenser, from a given quan- 
tity of heat." 

148. Harmonization of Statements. It has been asserted that the state- 
ments of the second law by different writers involve ideas so diverse as, 
apparently, not to cover a common principle. A moment's consideration 
of Art. 144 will explain this. The second law, in the forms given in (a), 
(b), (c), (g), is an axiom, from which the criterion of reversibility is estab- 
lished. In (d), (e) (/), it is a simple statement of the efficiency of the Car- 
not cycle, with which the axiom is associated ; while in (Ji), it is the 
criterion of reversibility itself. Confusion may be avoided by treating 
the algebraic expression of (d), Art. 146, as a sufficient statement of 
the second law, from which all necessary applications may be derived. 

149. Consequences of the Second Law. Some of these were touched upon in 
Art. 143. The first law teaches that heat and work are mutually convertible, 
the second law shows how much of either may be converted into the other under 
stated conditions. Ordinary condensing steam engines work between tempera- 
tures of about 350° F. and 100° F. The maximum possible efficiency of such 
engines is therefore 

350 - 100 



350 + 459.4 



= 0.31. 



72 



APPLIED THERMODYNAMICS 



The efficiencies of actual steam engines range from 2\ to 25 per cent, with an 
average probably not exceeding 7 to 10 per cent. A steam engine seems therefore 
a most inefficient machine ; but it must be remembered that, of the total heat 
supplied to it, a large proportion is (by the second law) unavailable for use, and 
must be rejected when its temperature falls to that of surrounding bodies. We can- 
not expect a water wheel located in the mountains to utilize all of the head of the 
water supply, measured down to sea level. The available head is limited by the 
elevation of the lowest of surrounding levels. The performance of a heat engine 
should be judged by its approach to the efficiency of the Carnot cycle, rather than 
by its absolute efficiency. 

Heat must be regarded as a "low unorganized'' form of energy, which pro- 
duces usef.ul work only by undergoing a fall of temperature. All other forms of 
energy tend to transform themselves into heat. As the universe slowly settles to 
thermal equilibrium, the performance of work by heat becomes impossible and all 
energy becomes permanently degenerated to its most unavailable form. 

150. Temperature Fall and Work Done. Consider the Carnot cycle, abed, 
Fig. 45, the total heat absorbed being -nabN and the efficiency abed -=- nabN 

= (T—t)-i-T. Draw the isothermals 
ef gh, ij, successively differing by equal 
temperature intervals ; and let the tem- 
peratures of these isothermals be T lt 
T 2 , T 3 . Then the work done in cycle 
abfe is nabNx(T- 7\)-f- T; that in 
cycle abhg is nabNx (T— T 2 )^T; that 
in cycle abji is nabN x(T— T 3 )-7-T. 
As (T-T,) = 3(T- Tj) and (T-T 2 ) 
= 2(7 7 -7 7 1 ), abji = 3(abfe) and abhg 
= 2 (abfe) ; whence abfe = efhg = ghji. 
In other words, the external work 
available from a definite temperature fall 
is the same at all parts of the thermometric scale. The waterfall analogy of 
Art. 149 may again be instructively utilized. 

151. Rankine's Statement of the Second Law. "If the total actual heat of a 
uniformly hot substance be conceived to be divided into any number of equal parts, the 
effects of those parts in causing work to be performed are equal.' 1 If we remember 
that by " total actual heat " Rankine means the heat corresponding to absolute tem- 
perature, his terse statement becomes a form of that just derived, dependent solely 
upon the computed efficiency of the Carnot cycle. 




Fig. 45. Arts. 150, 153, 154, 156.- 
Law of Thermodynamics 



•Second 



152. Absolute Temperature. It is convenient to review the steps by which 
the proposition of Art. 150 has been established. We have derived a conception 
of absolute temperature from the law of Charles, and have found that the effi- 
ciency of the Carnot cycle bears a certain relation to definite absolute temperatures. 



KELVIN'S ABSOLUTE SCALE 73 

Our scale of absolute temperatures, practically applied, is not entirely satisfactory ; 
for the absolute zero of the air thermometer, —459.4° F., is not a true absolute 
zero, because air is not a perfect gas. The logical scale of absolute temperature 
would be that in which temperatures were denned by reference to the work done 
by a reversible heat engine. Having this scale, we should be in a j>osition to com- 
pute the coefficient of expansion of a perfect gas. 

153. Kelvin's Scale of Absolute Temperature. Kelvin, in 1848, was led 
by a perusal of Carnot's memoir to propose such a scale. His first defini- 
tion, based on the caloric theory, resulted only in directing general atten- 
tion to Carnot's great work ; his second definition is now generally adopted. 
Its form is complex, but the conception involved is simply that of Art. 150: 

" The absolute temperatures of two bodies are proportional to the quanti- 
ties of heat respectively taken in and given out in localities at one temperature 
and at the other, respectively, by a material system subjected to a complete 
cycle of perfectly reversible thermodynamic operations, and not allowed to part 
with or take in heat at any other temperature." Briefly, 

" The absolute values of two temperatures are to each other in the propor- 
tion of the quantities of heat taken in and rejected in a perfect thermodynamic 
engine, working with a source and condenser at the higher and the lower of 
the temperatures respectively." Symbolically, 

T H . tj, . AK T nabN 

— = — ; or, in h is. 45, — = . 

t h t ndcN 

This relation may be obtained directly by a simple algebraic trans- 
formation of the equation for the second law, given in Art. 146, (d). 

154. Graphical Representation of Kelvin's Scale. Returning to Fig. 45, 
but ignoring the previous significance of the construction, let ab be an iso- 
thermal and an, bN adiabatics. Draw isothermals ef, gh, ij, such that the 
areas abfe, efhg, ghji are equal. Then if we designate the temperatures 
along ab, ef, gh, ij by T, T l} T. 2 , T s , the temperature intervals T— T lt 
T 1 - T 2 , T 2 - T 3 are equal. If we take ab as 212° F., and cd as 32° F., 
then by dividing the intervening area into 180 equal parts, we shall have 
a true Fahrenheit absolute scale. Continuing the equal divisions down 
below cd, we should reach a point at which the last remaining area be- 
tween the indefinitely extended adiabatics was just equal to the one next 
preceding, provided that the temperature 32°F. could be expressed in an 
even number of absolute degrees. 

155. Carnot's Function. Carnot did not find the definite formula for effi- 
ciency of his engine, given in Art. 135, although he expressed it as a function of 
the temperature range (T-t). We may state the efficiency as 

e = z(T-t), 



74 APPLIED THERMODYNAMICS 

z being a factor having the same value for all gases. Taking the general expres- 
sion for efficiency, H ~ h (Art. 128), and making H = h + dh, we have 
H 

h + dh - h dh 
e 



h + dh h + dh 
For e — z(T — t), we may write e = zdt or z =— , giving 

- off, equivalent to d 



h + dh hdt 

But — = — (Art. 153) ; whence — ■ — = — — — and — = — , and t = — = -. 
t h t h t h dh z 

\ y t T — t 

Then z =- and e = = in finite terms, as alreadv found. The factor z 

t t T J 

is known as CarnoVs function. It is the reciprocal of the absolute temperature. 

156. Determination of the Absolute Zero. The porous plug experiments con- 
ducted by Joule and Kelvin (Art. 74) consisted in forcing various gases slowly 
through an orifice. The fact has already been mentioned that when this action 
was conducted without the performance of external work, a barely noticeable 
change in temperature was observed ; this being with some gases an increase, and 
with others a decrease. When a resisting pressure was applied at the outlet of the 
orifice, so as to cause the performance of some external work during the flow of 
gas, a fall of temperature was observed ; and this fall ivas different for different gases. 

The " porous plug " was a wad of silk fibers placed in the orifice for the purpose 
of reconverting all energy of velocity back to heat. Assume a slight fall of tem- 
perature to occur in passing the plug, the velocity energy being reconverted to 
heat at the decreased temperature, giving the equivalent paths ad, dc, Fig. 45. 
Then expend a sufficient measured quantity of work to bring the substance back 
to its original condition a, along cha. By the second law, 

T T, . T, , T nrbN 

J- == 1 , and 



nabN nefN nabN — abfe" T x nabN — abfe' 

or T-T,= T, ( nahN -lW, $& 

1 x \nabN-abfe j l nabN - abfe 

If (T— 7\) as determined by the experiment = a, and nabN be put equal to unity, 

rp _ a(\ — abfe) 
abfe 

in which abfe is the work expended in bringing the gas back to its original tem- 
perature. This, in outline, was the Joule and Kelvin method for establishing a 
location for the true absolute zero: the complete theory is too extensive for pres- 
entation here (2). The absolute temperature of melting ice is on this scale 
491.58° F. or 273.1° C. 

The agreement with the hydrogen or the air thermometer is close. 
The correction for the former is generally less than t ±q° C, and that for 



THE SECOND LAW OF THERMODYNAMICS 75 

the latter less than T y C. Wood has computed (3) that the true absolute 
zero must necessarily be slightly lower than that of the air thermometer. 
According to Alexander, (4) the difference of the two scales is constant for 
all temperatures. The Kelvin absolute scale establishes a logical defini- 
tion of temperature as a physical unit. Actual gas thermometer tempera- 
tures may be reduced to the Kelvin scale as a final standard. 

In the further discussion, the temperature —459.6° F. will be regarded 
as the absolute zero. 

(1) Peabody, Thermodynamics, 1907, 27. (2) Phil. Trans., CXLIV, 349. 
(3) Thermodynamics, 1905, 116. (4) Treatise on Thermodynamics, 1892, 91. 



SYNOPSIS OF CHAPTER VII 

Statements of the second law : an axiom establishing the criterion of reversibility ; 
H — h __ T — t h _J L a statement of the efficiency of the Carnot cycle ; the cri- 

H TUT terion of reversibility itself. 

The second law limits the possible efficiency of a heat engine. 
The fall of temperature determines the amount of external work done. 
Temperature ratios defined as equal to ratios of heats absorbed and emitted. 
The Carnot function for cyclic efficiency is the reciprocal of the absolute temperature. 
The absolute zero, based on the secon 1 law, is at — 459.6° F. 



PROBLEMS 

1. Illustrate graphically the first and the second laws of thermodynamics. Frame 
a new statement of the latter. 

2. An engine works in a Carnot cycle between 400° F. and 280° F., developing 
120 h.p. If the heat rejected by this engine is received at the temperature of rejection 
by a second Carnot engine, which works down to 220° F. , find the horse power of the 
second engine. 

3. Find the coefficient of expansion at constant pressure of a perfect gas. What 
is the percentage difference between this coefficient and that for air ? 

4. A Carnot engine receives from the source 1000 B. t. u., and discharges to the 
condenser 500 B. t. u. If the temperature of the source is 600° F., what is the tem- 
perature of the condenser ? 

5. A Carnot engine receives from the source 190 B. t. u. at 1440.4° F., and dis- 
charges to the condenser 90 B. t. u. at 440.4° F. Find the location of the absolute 
zero. 

6. In the porous plug experiment, the initial temperature of the gas being that of 
melting ice, and the fall of temperature being T i ff of the range from melting to boiling 
of water at atmospheric pressure, the work expended in restoring the initial tempera- 
ture was 1.58 foot pounds. Find the absolute temperature at 32° F. 



76 APPLIED THERMODYNAMICS 



REVIEW PROBLEMS, CHAPTERS I-VII 

1. State the precise meaning, or the application, of the following expressions : 
k I y B I 778 P 

dT 



y 

pvloge — 

V 


pv -PV 
n-1 


pv - PV 

y-i 


pvv = c 


pv = c 


s -i n -y 



P\ T 

- = — pv n = c 

p j t n — 1 

2. A heat engine receives its fluid at 350° F. and discharges it at 110° F. It con- 
sumes 200 B. t. u. per Ihp. per minute. Find its efficiency as compared with that of 
the corresponding Carnot cycle. 

3. Given a cycle abc, in which P a = P b — 100 lb. per sq. in., V a = 1, 



V a 

PbVi, 1 -* = PcVc 1 - 8 , P a V a = P C V C , find the pressure, volume, and temperature at c if the 
substance is 1 lb. of air. 

4. Find the pressure of 100 lb. of air contained in a 100 cu -ft. tank at 82° F. 

5. A heat engine receives 1175.2 B. t. u. in each pound of steam and rejects 
1048.4 B. t. u. It uses 3110 lb. of steam per hour and develops 142 hp. Estimate the 
value of the mechanical equivalent of heat. 

6. One pound of air at 32° F. is compressed from 14.7 to 2000 lb. per square inch 
without change of temperature. Find the percentage change of volume. 

T — t 

7. Prove that the efficiency of the Carnot cycle is 

8. Air is heated at constant pressure from 32 c F. to 500° F. Find the percentage 
change in its specific volume. 

9. Prove that the change of internal energy in passing from a to & is independent 
of the path ab. 

10. Given the formula for change of internal energy, — - — — — - — -, prove that 
E h -E a = l{T h -T a ). y ~ l 

11. Given B for air = 53.36, V= 12.387 ; and given V= 178.8, k = 3.4 for hydro- 
gen : find the value of y for hydrogen. 

12. Explain isothermal, adiabatic, isodynamic, isodiabatic. 

13. Find the mean specific heat along the path pv 1 - 8 — c for air (/ = 0.1689). 

14. A steam engine discharging its exhaust at 212° F. receives steam containing 
1100 B. t. u. per pound at 500° F. What is the minimum weight of steam it may use 
per Ihp.-hr. ? 

15. A cycle is bounded by polytropic paths 12, 23, 13. We have given 

P 1= P 2 = 100,000 lbs. per sq. ft. 
F 2 = V s = 40 cubic feet per pound. 
T x = 3000° F. 
P X V X = P Z V Z . 
Find the amount of heat converted to work in the cycle. 



CHAPTER VIII 



ENTROPY 



157. Adiabatic Cycles. Let abdc, Fig. 46, be a Carnot cycle, an and bN 
the projected adiabatics. Draw intervening adiabatics em, gM, etc., so 
located that the areas naem, megM, MgbN, are equal. Then since the effi- 
ciency of each of the cycles aefc, eghf, gbdh, is (T — t) -t- T, the work areas 
represented by these cycles are all equal. To measure these areas by mechani- 
cal means would lead to approximate results only. 



158. Rectangular Diagram. If the adiabatics and isothermals 
were straight lines, simple arithmetic would suffice for the measure- 
ment of the work areas of Fig. 46. We 
have seen that in the Carnot cycle, 
bounded by isothermals and adiabatics, 



!=^ (Art. 153). 



Applying this for 



mula to Rankine's theorem (Art. 106), 
we have the quotient of an area and a 
length as a constant. If the area h is 
a part of IT, then there must be some 
constant property, which, when multi- 
plied by the temperatures T or t, will 




Fig. 46. Arts. 157, 158, 159, 160. 
Adiabatic Cycles. 




Fig. 47. 



Arts. 158, 163, 171. 
tropy Diagram. 



give the areas IT or h. Let us conceive 
of a diagram in which only one coor- 
dinate will at present be named. That 
coordinate is to be absolute temperature. 
Instead of specifying the other coordi- 
nate, let it be assumed that subtended 
areas on this diagram are to denote 
quantities of heat absorbed or emitted, 
just as such areas on the PV diagram 
represent external work done. As an 
example of such a diagram, consider 
Fig. 47. Let the substance be one 
77 



78 



APPLIED THERMODYNAMICS 



pound of water, initially at a temperature of 32° F., or 491.6° abso- 
lute, represented by the height ab, the horizontal location of the 
state b being taken at random. Now assume the water to be heated 
to 212° F., or 671.6° absolute, the specific heat being taken as con- 
stant and equal to unity. The heat gained is 180 B. t. u. The 
final temperature of the water fixes the vertical location of the 
new state point c?, i.e. the length of the line cd. Its horizontal lo- 
cation is fixed by the consideration that the area subtended between 
the path bd and the axis which we have marked ON shall be 
180 B. t. u. The horizontal distance ac may be computed from the 
properties of the trapezoid abdc to be equal to the area abdc divided 
by [(ab + cd) -r- 2] or to 180 h- [(491.6 + 671.6) -r- 2] = 0.31. The 
point d is thus located (Art. 163). 

159. Application to a Carnot Cycle. Ordinates being absolute 
temperatures, and areas subtended being quantities of heat absorbed 
or emitted, we may conclude that an isothermal must be a straight 
horizontal line; its temperature is constant, and a finite amount of 
heat is transferred. If the path is from left to right, heat is to be 

conceived as absorbed; if from right to 
left, heat is rejected. Along a (re- 
versible) adiabatic, no movement of heat 
occurs. The only line on this diagram 
which does not subtend a finite area is 
a straight vertical line. Adiabatics are 
consequently vertical stra'ght lines. (But 
see Art. 176.) The temperature must 
constantly change along an adiabatic. 
The lengths of all isothermals drawn be- 
tween the same two adiabatics are equal. 
The Carnot cycle on this new diagram 
may then be represented as a rectangle enclosed by vertical and hori- 
zontal lines ; and in Fig. 48 we have a new illustration of the cycles 
shown in Fig. 46, all of the lines corresponding. 



T 


c 


\, 

f 


h 


d 




n 


m 


M 


N 


N 



Fig. 48. Arts. 159, 160, 161, 163, 
166. — Adiabatic Cycles, Entropy- 
Diagram. 



160. Physical Significance. The new diagram is to be conceived 
as so related^ to the P V diagram of Fig. 46 that while an imaginary 



ENTROPY 



79 



pencil is describing any stated path on the latter, a corresponding 
pencil is tracing its path on the former. In the PV diagram, the 
subtended areas constantly represent external work done by or on the 
substance ; in the new diagram they represent quantities of heat ab- 
sorbed or rejected. (Note, however, Art. 176.) The area of the 
closed cycle in the first case represents the net quantity of work done; 
in the second, it represents the net amount of heat lost, and conse- 
quently, also, the net work done. But subtended areas under a single 
path on the PV diagram do not represent heat quantities, nor in the 
new diagram do they represent work quantities. The validity of the 
diagram is conditioned upon the absoluteness of the properties chosen as 
coordinates. We have seen that temperature is a cardinal property, 
irrespective of the previous history of the substance ; and it will be 
shown that this is true also of the horizontal coordinate, so that we 
may legitimately employ a diagram in which these two properties 
are the coordinates. 



161. Polytropic Paths. For any path in which the specific heat 
is zero, the transfer of heat is zero, and the path on this diagram is 
consequently vertical, an adiabatic. For specific heat equal to 
infinity, the temperature 
cannot change, and the 
path is horizontal, an iso- 
thermal. For any positive 
value of the specific heat, 
heat area and temperature 
will be gained or lost 
simultaneously ; the path 
will be similar to ai or aj, 
Fig. 48. If the specific 
heat is negative, the tem- 
perature will increase with 
rejection of heat, or de- 



T 1 

.oV>/ CO 

/ s 

f Y n = 1 s = ^ ° 


\ 

Is n = l s = qc £ ' 


>^ y " 

*/ s 
<?/ «-; 
/ " 

< 


N 



Fig. 49. 



Arts. 161, 165. — Polytropic Paths on 
Entropy Diagram. 



crease with its absorption, as along the paths ak, al, Fig. 48. These 
results may be compared with those of Art. 115. Figure 49 shows 
on the new diagram the paths corresponding with those of Fig. 31. 
It may be noted that, in general, though not invariably, increases of 



80 



APPLIED THERMODYNAMICS 



volume are associated with increases of the horizontal coordinate of 
the new diagram. 

162. Justification of the Diagram. In the PV diagram of Fig. 50, consider 
the cycle ABCD. Let the heat absorbed along a portion of this cycle be repre- 
sented by the infinitesimal strips nabN, 
NbcM, Mcdm, formed by the indefinitely 
projected adiabatics. In any one of these 
strips, as nabN, we have, in finite terms, 

nabN T nabN neoN 

= — , or = — ■- — . 

negN t T t 

Considering the whole series of strips 
from A to C, we have 




nabN 






negN 



Fig. 50. Art. 162. — Entropy a Cardinal 
Property. 



or, using the symbol H for heat trans- 
ferred, 

dH 



2-1 T 



o, 



in which T expresses temperature generally. 

Let the substance complete the cycle ABC DA; we then have, the paths being 
reversible, 



f dH _ 1 dH I 

\ 7 J; ^ + J: 



dH 



0; 



while for the cycle ADCDA, 



whence, 



: f- . f ♦ • 

A %Ja KJg 

Jic pc 

'1H - I IK 
D T ~ I 3 T ' 



dH 
T 



0: 



The integral f ® thus has the same value whether the path is A DC or ABC, 

or, indeed, any reversible path between A and C ; its value is independent of the 
path of the substance. Now this integral will be shown immediately to be the most 
general expression for the horizontal coordinate of the diagram under discussion. 
Since it denotes a cardinal property, like pressure or temperature, it is permissible 

to use a diagram in which the coordinates are T and j — — 



ENTROPY 



81 



163. Analytical Expression. Along any path of constant tem- 
perature, as ab, Fig. 48, the horizontal distance nN may be computed 
from the expression, nN= H '-r- T, in which H represents the quan- 
tity of heat absorbed, and T the temperature of the isothermal. If 
the temperature varies, the horizontal component of the path during 
the absorption of dH units of heat is dn = dH-h T. For any path 
along which the specific heat is constant, and equals Jc, JcdT= dH, 



j kdT , 
an = — — , ana n 



Jt T 



h loo-. — • 
T Jt T * e t 

If the specific heat is variable, say k — a + bT, then 

The line bd of Fig. 47 is then a logarithmic curve, not a straight 
line ; and the method of finding it adopted in Art. 158 is strictly 
accurate only for an infinitesimal change of temperature. Writing 
the expression just derived in the form n = klog e (T-r- 1) and remem- 
bering that T= PV-s- R, while t=*pv-*- B, we have 

n = k log e (P V-z- pv~) . 
The expression Jclog e (T-v-t~) is the one most 
commonly used for calculating values of the hori- 
zontal coordinate for polytropic paths. The 
expression dn = dH-s- T is general for all re- 
versible paths and is regarded by Rankin e as 
the fundamental equation of thermodynamics. 




Fig. 51. Art. 164.— Graphi- 
cal Determination of 
Specific Heat. 



164. Computation of Specific Heat. If at any 

point on a reversible path a tangent be drawn, the 

length of the subtangent on the iV-axis represents the 
value of the specific heat at 

that point. In Fig. 51, draw the tangent nm to the 
curve AB at the point wand construct the infinitesimal 
triangle dtdn. From similar triangles, mr : nr : : dn : dt, 
or mr = Tdn ~- dt = dH ~ dt = k (Art. 58). 



165. Comparison of Specific Heats. If a gas is 

heated at constant pressure from a. Fig. 52, it will 
gain heat and temperature, following some such 
path as ab. If heated at constant volume, 
through an equal range of temperature, a less 




d f c 

Fig. 52. Art. 165. — Com- 
parison of Specific Heats. 



82 APPLIED THERMODYNAMICS 

quantity of heat will be gained ; i.e. the subtended area aefd will be less 
than the area abed. In general, the less the specific heat, the more 
nearly vertical will be the path. (Compare Fig. 49.) When k = 0, the 
path is vertical ; when k = oo, the path is horizontal. 

166. Properties of the Carnot Cycle. In Fig. 48, it is easy to see that 
since efficiency is equal to net expenditure of heat divided by gross ex- 
penditure, the ratio of the areas abdc and abNn expresses the efficiency, 
and that this ratio is equal to ( T — t ) -r- T. The cycle abdc is obviously 
the most efficient of all that can be inscribed between the limiting iso- 
thermals and adiabatics. 

167. Other Deductions. The net enclosed area on the TN diagram 
represents the net movement of heat. That this area is always equivalent 
to the corresponding enclosed area on the PV diagram is a statement of 
the first law of thermodynamics. Two statements of the second law have' 
just been derived (Art. 166). The theorem of Art. 106, relating to the 
representation of heat absorbed by the area between the adiabatics, is 
accepted upon inspection of the TN diagram. That of Art. 150, from which 
the Kelvin absolute scale of temperature was deduced, is equally obvious. 

168. Entropy. The horizontal or N coordinate on the diagram 
now presented was called by Clausius the entropy of the body. It 

may be mathematically defined as the ratio n= ( —— • The physical 

definition or conception should be framed by each reader for himself. 
Wood calls entropy " that property of the substance which remains 
constant throughout the changes represented by a [reversible] adia- 
batic line." It is for this reason that reversible adiabatics are called 
isentropics, and that we have used the letters n, N in denoting 
v adiabatics. 



^^ 169. General Formulas. It must be thoroughly 

/ ^> v 5 understood that the validity of the entropy diagram is 

j ; \. dependent upon the fact that entropy is a cardinal prop- 

-i-c -"^z erty, like pressure, volume, and temperature. For this 

reason it is desirable to become familiar with compu- 
tations of change of entropy irrespective of the path 

a/ pursued. Otherwise, the method of Art. 163 is usually 

Fig. 54. Art. 169. — Change more convenient. 

of Entropy. Consider the states a and b, Fig. 54. Let the 

substance pass at constant pressure to c and thence at constant volume 



ENTROPY 83 

T T 

to b. The entropy increases by k log e — c - +1 log e — (Art. 163), k and I 

-*■ a ■*■ c 

in this instance denoting the respective special values of the specific 
heats. An equivalent expression arises from Charles' law : 

» = * l»g« ~ + 1 log, % = ft log, I? + I log, 5, (A) 

in which last the final and initial states only are included. 
We may also write, 

Yi 

T I*tV* log v 
n=l log c y+ / - / y 8e % Arts. 94, 95, 163, 

= 2 log, ^ +■(* - log e y\ Arts. 51, 65 : (B) 

and further, 

7i = fclog e ^ + (^-01og e ^ 

= * l0ge |?+(*- Q lOg. ^ ' (C) 

The entropy is completely determined by the adiabatic through the state point. 
In the expression n 1 = A; 1 log e — , the value of n L apparently depends upon that of Jc v 

which is of course related to the path; along another path, the gain or loss of 

T 
entropy might be n 2 = k 2 log e — > a different value ; but although the temperatures 

would be the same at the beginning and end of both processes, the pressures or 
volumes would differ. The states would consequently be different, and the values 
of n should therefore differ also. 

A graphical method for the transfer of perfect gas paths from the PFto the 
TN plane has been developed by Berry (1). 

170. Other Names for n. Rankine called n the thermodynamic func- 
tion. It has been called the " heat factor." Zeuner describes it as " heat 
weight." It has also been called the " mass " of heat. The letters T, JV, 
•which we have used in marking the coordinates, were formerly replaced 
by the Greek letters theta and phi, indicating absolute temperatures and 
entropies ; whence the name, theta-phi diagram. The TN diagram is now 
commonly called the temperature-entropy diagram, or, more briefly, the 
entropy diagram. 

171. Entropy Units. Thus far, entropy has been considered as a hori- 
zontal distance on the diagram, without reference to any particular zero 
point. Its units are B. t. u. per degree of absolute temperature. Changes 



84 APPLIED THERMODYNAMICS 

of entropy are alone of physical significance. The choice of a zero point 
may be made at random ; there is no logical zero of entropy. A conven- 
ient starting point is to take the adiabatic of the substance through the 
state P = 2116.8, T = 32° F., as the OT axis, the entropy of this adiabatic 
being assumed to be zero, as in ordinary tables. Thus, in Fig. 47 (Art. 
158), the OT axis should be shifted to pass through the point b, which 
was located at random horizontally. 

172. Hydraulic Analogy. The analogy of Art. 140 may be extended to illus- 
trate the conception of entropy. Suppose a certain weight of water W to be 
maintained at a height H above sea level ; and that in passing through a motor 
its level is reduced to h. The initial potential energy of the water may be 
regarded as WH; the final residual energy as Wh\ the energy expended as 
W(H — 7i). Let this operation be compared with that of a Carnot cycle, taking 
in heat at T and discharging it at t. Regarding heat as the product of N and T, 
then the heat energies corresponding to the water energies just described are NT, 
Nt, and N(T — t) ; N being analagous to W, the weight of the water. 

173. Adiabatic Equation. Consider the states 1 and 2, on an adiabatic 
path, Fig. 55. The change of entropy along the constant volume path 13 

P f 

is I log e — ; that along the constant pressure path 32 

X ^T 

jVo, is k log c — ■ The difference of entropy between 

• %. 1 and 2, irrespective of the path, is 

I log e ^ + k log e -- = I log e — 2 + k log e — 2 - 



Fig. 55.- Art, 173.— ^ or a reversible adiabatic process, this is equal to 
Adiabatic Equation. zero ; whence 

I log e §= - k log e ^ 2 , or y log e V 2 + log e P 2 = y log e V x + log^ 

from which we readily derive P x Vf = P-tVl, the equation of the adiabatic. 

174. Use of the Entropy Diagram. The intelligent use of the entropy 
diagram is of fundamental importance in simplifying thermodynamic con- 
ceptions. The mathematical processes formerly adhered to in presenting 
the subject have been largely abandoned in recent text-books, largely on 
account of the simplicity of illustration made possible by employing the 
TN coordinates. 

Belpaire was probably the first to appreciate their usefulness. Gibbs, at about 
the same date, 1873, presented the method in this country and first employed as 
coordinates the three properties volume, entropy, and internal energy. Linde, 



IRREVERSIBLE PROCESSES 



85 



Schroeter, Hermann, Zeuner, and Gray used TN diagrams prior to 1890. Cotterill, 
Ayrton and Perry, Dwelshauvers Dery and Ewing have employed them to a con- 
siderable extent. Detailed treatments of this thermodynamic method have been 
given by Boulvin, Reeve, Berry, and Golding (2). Some precautions necessary in 
its practical application are suggested in Arts. 454-458. 



Irreversible Processes 



It is of importance to distinguish 
in relation to entropy changes. 



175. Modification of the Entropy Conception, 
between reversible and irreversible processes 
The significance of the term reversible, as ap- 
plied to a path, was discussed in Art. 125. A 
process is reversible only when it consists of a 
series of successive states of thermal equilib- 
rium. A series of paths constitute a reversible 
process only when they form a closed cycle, 
each path of which is itself reversible. The 
Carnot cycle is a perfect example of a reversible 
process. As an example of an irreversible cycle, 
let the substance, after isothermal expansion, 
as in the Carnot cycle, be transferred directly 
to the condenser. Heat will be abstracted, and 
the pressure may be reduced at constant vol- 
ume, as along be, Fig. 56. Then allow it to compress isothermally, as in the 
Carnot cycle, and finally to be transferred to the source, where the temperature 
and pressure increase at constant volume, as along da. This cycle cannot be 
operated in the reverse order, for the pressure and temperature cannot be reduced 
from a to d while the substance is in communication with the source, nor increased 
from c to b while it is in communication with the condenser. 




^°THEB MAL 



Fig. 56. 



Art. 175.- 

Cycle. 



Irreversible 



176. Irreversibility in the Porous Plug Experiment. We have seen that in this 
instance of unresisted expansion, the fundamental formula of Art. 12 becomes 
H = T + I +W + V (Art. 127). Knowing H = 0, W = 0, we may write 
(7 T + I) = — V, or velocity is attained at the expense of the internal energy. The 
velocity evidences kinetic energy ; mechanical work is made possible ; and we might 
expect an exhibition of such work and a fall of internal energy, and consequently 
of temperature. But we find no such utilization of the kinetic energy of the rapidly 
flowing jet; on the contrary, the gas is gradually brought to rest and the velocity 
derived from an expenditure of internal energy is reconverted to internal energy. 
The process was adiabatic, for no transfer of heat occurred ; it was at the same 
time isothermal, for no change of temperature occurred ; and while both adiabatic 
and isothermal, no external work was done, so that the PV diagram is invalid. 

Further : the adiabatic path here considered w^as not isentropic, like an ordinary 

adiabatic. The area under the path on the TN diagram no longer represents heat 

dJ-f 
absorbed from surrounding bodies. Neither does dn = — , for H is zero, while 



86 APPLIED THERMODYNAMICS 

n is finite. The expression for increase of entropy, f , along a reversible path, does 
not hold for irreversible operations. 

In irreversible operations, the expression \ — - ceases to represent a cardinal 

property. We have the following propositions : 

(a) In a reversible operation, the sum of the entropies of the participating substances 
is unchanged. During a reversible change, the temperatures of the heat-absorbing 
and heat-emitting bodies must differ to an infinitesimal extent only; they are in 
finite terms equal. The heat lost by the one body is equal to the heat gained by 

the other, so that the expression I — denotes both the loss of entropy by the one 

substance and the gain by the other; the total stock of entropy remaining constant. 
(ft) During irreversible operations, the aggregate entropy increases. Consider two 
engines working in the Carnot cycle, the first taking the quantity of heat H 1 from 
the source, and discharging the quantity H 2 to the condenser ; the second, acting 
as a heat pump (Art. 139), taking the quantity H 2 ' from the condenser and restoring 
HJ to the source. Then if the work produced by the engine is expended in driving 
the pump, without loss by friction, 

H x — H 2 — H^ — H 2 . 

If the engine is irreversible, H^> H^, or H l — H x '>0, whence, H 2 — i7 2 '>0. If 
we denote by a a positive finite value, H 1 = H^ + a and H 2 = H 2 + a. But 
H{ T x H Y ' H> ■ ' 

jp=Y ,or T ~T = ' -consequently 

H x -a H 2 -a ' , H, H ( 1 1 \ 



Since 7\ > T 2 , -^ - -=? < 0, or -^> 7 =±, or, generally, f — < 0. The value of 



'dH 



C — is thus, for irreversible operations, negative. 

Now let a substance work irreversibly from A to B, thence reversibly from B to 
A. We may write 

f¥ + rf=ff-rf<»- 

(irrevO (rev.) (irrev.) (rev.) 

J' %B dH 
-— - , 
a T 

dH being the amount of heat absorbed along any reversible path, while the change 

of entropy of the source which supplies the substance with heat (reversibly) is 

r B dH 
Njj — N a = — I - — , the negative sign denoting that heat has been abstracted. 

Ja t 
We have then, from equation (A), 

- (AY - N A >) - (N B - N A ) < ; or, (N B + N B ') - (N A + N A ') >0; 

i.e. the sum of the entropies of the participating substances increases when the 
process is irreversible. 



.f 



IRREVERSIBILITY 87 

(c) The loss of work due to irreversibility is proportional to the increase of entropy. 
Consider a partially completed cycle : one which might be made complete were all 
of the paths reversible. Let the heat absorbed be Q, at the temperature T, in- 
creasing the entropy of the substance by ^; and let its entropy be further increased 
by N' — N during the process. The total increase of entropy is then n — N' — JV+ — , 

whence Q = T(n — N' + N). The work done -may be written as H — H' + Q, in 
which H and H' are the initial and final heat contents respectively. Calling this 

W, we have 

W = II - H' + T(n - N' + N). 

In a reversible cycle j — = n — ; whence W R = H — II' 4- T^iV — iV 7 ) and 
IF* - TF = Tn. T 

(A careful distinction should be made at this point between the expression 

7 TT 

and the term entropy. The former is merely an expression for the latter 

under specific conditions. In the typical irreversible process furnished by the 
porous plug experiment, the entropy increased; and this is the case generally with 

such processes, in which dn>—— • Internal transfers of heat may augment the 

entropy even of a heat-insulated body, if it be not in uniform thermal condition. 
Perhaps the most general statement possible for the second law of thermody- 
namics is that all actual processes tend to increase the entropy; as we have seen, this 
keeps possible efficiencies below those of the perfect reversible engine. The prod- 
uct of the increase of entropy by the temperature is a measure of the waste of 
energy (3).) 

Most operations in power machinery may without serious error be analyzed 
as if reversible ; unrestricted expansions must always be excepted. The entropy 
diagram to this extent ceases to have " an automatic meaning." 

(1) The Temperature-Entropy Diagram, 1908. (2) See Berry, op. cit. (3) The 
works of Preston, Swinburne, and Planck may be consulted by those interested in this 
aspect of the subject. 



SYNOPSIS OF CHAPTER VIII 

It is impracticable to measure PFheat areas between the adiabatics. 

The rectangular diagram : ordinates = temperature ; areas = heat transfers. 

Application to a Carnot cycle : a rectangle. 

The validity of the diagram is conditioned upon the absoluteness of the horizontal 

coordinate. 
The slope of a path of constant specific heat depends upon the value of the specific heat. 

The expression i - — has a definite value for any reversible change of condition, 

regardless of the path pursued to effect the change. 

dll T T 

dn = — , or n = k log e — for constant specific heat = k, or n = a log e — -f- b(T— t) for 

variable specific heat = a + bT. 



88 APPLIED THERMODYNAMICS 

The value of the specific heat along a poly tropic is represented by the length of the sab- 
tangent. 

Illustrations : comparison of k and I ; efficiency of Carnot cycle ; the first law ; the 
second law ; heat area between adiabatics ; Kelvin's absolute scale. 

Entropy units are B. t. u.per degree absolute. The adiabatic for zero entropy is at 
T=3£°F., P= 2116.8. 

1l=k l0g e -^ + I l0 ge ^ = I l0ge^ + (* - Z)l0g e |* = k l0g e |A + (fc _ ?)log e ^ . 
y<t Pa la Va la Pb 

Hydraulic analogy ; physical significance of entropy ; use of the diagram. 
Derivation of the adiabatic equation. 

Irreversible Processes 

A reversible cycle is composed of reversible paths ; example of an irreversible cycle. 
Joule's experiment as an example of irreversible operation. 

Heat generated by mechanical friction of particles ; the path both isothermal and adia- 
batic, but not isentropic. 

H= T+I+ W+ For V = -(I+T). 

(I FT 

For irreversible processes, dn is not equal to - — ; the subtended area does not repre- 
sent a transfer of heat ; non-isentropic adiabatics. 

In reversible operations, the aggregate entropy of the participating substances is 
unchanged. 

J ,7 TT 
- — <o. 

The loss of work due to increase of entropy is n T ; dn > §JL. 

T 



PROBLEMS 

1. Plot to scale the TJVpath of one pound of air heated (a) at constant pressure 
from 100° F. to 200° F., then (b) at constant volume to 300° F. The logarithmic 
curves may be treated as two straight lines. 

2. Construct the entropy diagram for a Carnot cycle for one pound of air in which 
T= 400° F., t = 100° F., and the volume in the first stage increases from 1 to 4 cubic 
feet. Do not use the formulas in Art. 169. 

3. Plot on the TN diagram paths along which the specific heats are respectively 
0, oo, 3.4, 0.23, 0.17, -0.3, -10.4, between T = 459.6 and T- 919.2, treating the 
logarithmic curves as straight lines. 

4. The variable specific heat being 0.20-0.0004 T- 0.000002 T 2 (T being in 
Fahrenheit degrees), plot the TN path from 100° F. to 140° F. in four steps, using 
mean values for the specific heat in each step. 

Find by integration the change of entropy during the whole operation. 

5. What is the specific heat at T = 40 (absolute) for a path the equation of 
which on the TN diagram is TN ' = c = 1200 ? 

6. Confirm Art. 134 by computation from the TN diagram. 

7. Plot the path along which 1 unit of entropy is gained per 100° absolute. What 
is the mean specific heat along this path from 0° to 400° absolute ? 



ENTROPY 89 

8. What is the entropy measured above the arbitrary zero per pound of air at 
normal atmospheric pressure in a room at 70° F. ? 

9. Find the arbitrary entropy of a pound of air in the cylinder of a compressor at 
2000 lb. pressure per square inch and 142° F. 

10. Find the entropy of a sphere of hydrogen 10 miles in diameter at atmospheric 
pressure and 175° F. 

11. The specific heat being 0.24 4- 0.0002 T, find the increase in entropy between 
459.6 and 919.2 degrees, all absolute. What is the mean specific heat over this range ? 



CHAPTER IX 

COMPRESSED AIR 

177. Compressed Air Engines. Engines are sometimes used in which the 
working substance is cold air at high pressure. The pressure is previously pro- 
duced by a separate device ; the air is then transmitted to the engine, the latter 
being occasionally in the form of an ordinary steam engine. This type of motor 
is often used in mines, on locomotives, or elsewhere where excessive losses by con- 
densation would follow the use of steam. For small powers, a simple form of 
rotary engine is sometimes employed, on account of its convenience, and in spite 
of its low efficiency. The absence of heat, leakage, danger, noise, and odor makes 
these motors popular in those cities where the public distribution of compressed 
air from central stations is practiced (1). The exhausted air aids in ventilating 
the rooms in which it is used. 

178. Other Uses of Compressed Air. Aside from the driving of engines, high- 
pressure air is used for a variety of purposes in mines, quarries, and manufac- 
turing plants, for operating hoists, forging and bending machines, punches, etc., 
for cleaning buildings, for operating " steam " hammers, and for pumping water 
by the ingenious "air lift" system. In many works, the amount of power trans- 
mitted by this medium exceeds that conveyed by belt and shaft or electric wire. 
The air is usually compressed by steam power, and it is obvious that a loss must 
occur in the transformation. This loss may be offset by the convenience and ease 
of transmitting air as compared with steam ; the economical generation, distribu- 
tion, and utilization of this form of power have become matters of first importance. 

The first applications were made during the building of the Mont Cenis tun- 
nel through the Alps, about 1860 (2). Air was there employed for operating 
locomotives and rock drills, following Colladon's mathematical computation of 
the small loss of pressure during comparatively long transmissions. A general 
introduction in mining operations followed. Two-stage compressors with inter- 
coolers were in use in this country as early as 1881. Among the projects sub- 
mitted to the international commission for the utilization of the power of Niagara, 
there were three in which distribution by compressed air was contemplated. Wide- 
spread industrial applications of this medium have accompanied the perfecting of 
the small modern interchangeable "pneumatic tools." 

179. Air Machines in General. In the type of machinery under consideration, 
a considerable elevation of pressure is attained. Centrifugal fans or paddle-wheel 
blowers, commonly employed in ventilating plants, move large volumes of air at 
very slight pressures, — usually a fraction of a pound, — and the thermodynamic 

90 



THE AIR ENGINE 



91 



relations are unimportant. Rotary blowers are used for moderate pressures, — up 
to 20 lb., — but they are generally wasteful of power and are principally employed 
to furnish blast for foundry cupolas, forges, etc. The machine used for com- 
pressing air for power purposes is ordinarily a piston compressor, mechanically 
quite similar to a reciprocating steam engine. These compressors are sometimes 
employed for comparatively low pressures also, as "blowing engines." 



The Air Engine 

180. Air Engine Cycle. In Fig. 57, ABOB represents an ideal- 
ized air engine cycle. AB shows the admission of air to the cylin- 
der. Since the latter is small as compared with the transmitting 
pipe line, the specific volume and pres- 
sure of the fluid, and consequently 
its temperature as well, remain un- 
changed. BO represents expansion 
after the supply from the mains is 
cut off. If the temperature at B is that 
of the external atmosphere, and ex- 
pansion proceeds slowly, so that any 
fall of temperature along BC is offset 
by the transmission of heat from the 
outside air through the cylinder walls, 
this line is an isothermal. If, however, 
expansion is rapid, so that no transfer 

of heat occurs, BO will be an adiabatic. In practice, the expansion 
line is a poly tropic, lying usually between the adiabatic and the 
isothermal. OB represents the expulsion of the air from the cyl- 
inder at the completion of the working stroke. At B, the inlet 
valve opens and the pressure rises to that at A. The volumes 
shown on this diagram are not specific volumes, but volumes of air in 
the cylinder. Subtended areas, nevertheless, represent external work. 




Fig. 57. Arts. 180-183, 189, 222, 223, 
226, Prob. 6.— Air Engine Cycles. 



181. Modified Cycle. The additional work area LMO obtained by ex- 
pansion beyond some limiting volume, say that along xy, is small. A 
slight gain in efficiency is thus made at the cost of a much larger cylin- 
der. In practice, the cycle is usually terminated prior to complete expan- 
sion, and has the form ABLMD, the line LM representing the fall of 
pressure which occurs when the exhaust valve opens. 



92 APPLIED THERMODYNAMICS 

182. Work Done. Letting p denote the pressure along AB, P 
the pressure at the end of the expansion, q the " back pressure " 
along MD (slightly above that of the atmosphere), and letting v 
denote the volume at B, and F"that at the end of expansion, both 
volumes being measured from OA as a line of zero volumes, then, 
for isothermal expansion, the work done is 

y 

pv -f pv log e q V; 

and for expansion such that pv n = P V n , it is 

, pv- PV ir 

pv + £ q V. 

n — 1 

183. Maximum Work. Under the most favorable conditions, expan- 
sion would be isothermal and " complete " ; i.e. continued down to the 
back-pressure line CD. Then, q = P = pv-±- V, and the work would be 
pv log e (F-7- v). For complete adiabatic expansion, the work would be 

pv + pv^PV_ PVH PV) (_y_\ 

y-l V2/-V 

184. Entropy Diagram. This cannot be obtained by direct transfer from the 
PV diagram, because we are dealing with a varying quantity of air. The method 
of deriving an illustrative entropy diagram is explained in Art. 218. 

185. Fall of Temperature. If air is received by an engine at 
P, T, and expanded to p, t, then from Art. 104, if P -±p— 10, and 
T= 529° absolute, with adiabatic expansion, t= — 187° F. 

This fall of temperature during adiabatic expansion is a serious matter. 
Low final temperatures are fatal to successful working if the slightest 
trace of moisture is present in the air, on account of the formation of ice 
in the exhaust valves and passages. This difficulty is counteracted in 
various ways: by circulating warm air about the exhaust passages; by 
specially designed exhaust ports; by a reduced range of pressures; by 
avoidance of adiabatic expansion (Art. 219) ; and by thoroughly drying 
the air. The jacketing of the cylinder with hot air has been proposed. 
Unwin mentions (3) the use of a spray of water, injected into the air 
while passing through a preheater (Art. 186). This reaches the engine 
as steam and condenses during expansion, giving up its latent heat of 
vaporization and thus raising the temperature. In the experiments on 
the use of compressed air for street railway traction in New York, stored 
hot water was employed to preheat the air. The only commercially sue- 



PREHEATERS 



93 



cessful method of avoiding inconveniently low temperatures after expan- 
sion is by raising the temperature of the inlet air. 



186. Preheaters. In the Paris installation (4), small heaters were 
placed at the various engines. These were double cylindrical boxes of 
cast iron, with an intervening space through which the air passed in a 
circuitous manner. The inner space contained a coke fire, from which 
the products of combustion passed over the top and down the outside of 
the outer shell. For a 10-hp. engine, the extreme dimensions of the 
heater were 21 in. in diameter and 33 in. in height. 

187. Economy of Preheaters. The heat used to produce elevation of 
temperature is not wasted. The volume of the air is increased, and the 
weight consumed in the 
engine is correspondingly 
decreased. Kennedy esti- 
mated in one case that 
the reduction in air con- 
sumption due to the in- 
crease of volume should 
have been, theoretically, 
0.30; actually, it was 0.25. 
The mechanical efficiency 
(Art. 214) of the engine 
is improved by the use of 
preheated air. In 
one instance, Ken- 
nedy computed a 
saving of 225 cu. ft. of 
"free" air (i.e. air at at- 
mospheric pressure and tem- 
perature) to have been ef- 
fected at an expenditure 
of 0.4 lb. of coke. Unwin 
found that all of the air 
used by a 72-hp. engine 
could be heated to 300° P. 
by 15 lb. of coke per hour. 
Figure 58 represents a 
modern form of preheater. Fig. 58. Art. 187.— Rand Air Preheater. 




188. Volume of Cylinder. If n be the number of single strokes per 
minute of a double-acting engine, Fthe cylinder volume (maximum vol- 



94 APPLIED THERMODYNAMICS 

ume of fluid), IT the number of pounds of air used per minute, v the specific 
volume of the air at its lowest pressure p and its temperature t, j^the 
horse power of the engine, and U the work done in foot-pounds per pound 
of air, then, ignoring clearance (the space between the piston and the cyl- 
inder head at the end of the stroke), the volume swept through by the 

piston per minute = Wv = 2 nV= WR-; 

P 

whence V= -^; and since WU = 33,000 N. JT= 33000 N , 
2np U 

, 1T 33000 NRt 
and V= ' 

2nUp 

189. Compressive Cycle. For quiet running, as well as for other 
reasons, to be discussed later, it is desirable to arrange the valve 
movements so that some air is gradually compressed into the clear- 
ance space during the latter part of the return stroke, as along Ua, 
Fig. 57. This is accomplished by causing the exhaust valve to close 
at E, the inlet valve opening at a. The work expended in this com- 
pression is partially recovered during the subsequent forward stroke, 
the air in the clearance space acting as an elastic cushion. 

190. Actual Design. A single-acting 10-hp. air engine at 100 r. p. m., 
working between 114.7 and 14.7 lb. absolute pressure, with an " appar- 
ent " (Art. 450) volume ratio during expansion of 5 : 1 and clearance equal 
to 5 per cent of the piston displacement, begins to compress when the 
return stroke of the piston is t 9 -q completed. The expansion and compres- 
sion curves are PV 13 = c. Assuming that the actual engine will give 90 
per cent of the work theoretically computed, find the size of cylinder 
(diameter == stroke) and the free air consumption per Ihp.-hr. 

In Fig. 59, draw the lines ab and cd representing the pressure limits. We are 
to construct the ideal PV diagram, making its enclosed length represent, to any 
convenient scale, the displacement of the piston per stroke. The extreme length 
of the diagram from the oP axis will be 5 per cent greater, on account of clear- 
ance. The limiting volume lines ef and gli are thus sketched in ; BC is plotted, 

making -^- = 5 ; the point E is taken so that = 0.9, and EF drawn. Then 

AB Di 

ABCDEF is the ideal diagram. We have, putting Di = D, 

P A = P B = 1U.7. 
P D =P E = U.7. 
V C =V D = 1MD. 
V £ = 0.2oD. 
V F = F A = 0.05 2). 
V E = 0.15 2). 



DESIGN OF AIR ENGINE 



95 



P G V n =P B V B » or P = PJ^rf = U^jff)" = 17 ' 75 ' 

PwVf = P E V E " or P F = P E i^f = 14,7 (fJi) 13 = 6L3L 
Work per stroke =jABi + iBCm - EDmk -jFEk 
= P A (V B - V a ) + 



Pb Vv-PcVc p " v v , PwVf-Pe Ve 



n-1 



= 14*[(114.7 x 0.20 D) + (U4-7 x0.2BII)- 8 (17.75x 1.05 J) 



- (14.7 x 0.9 Z>) 
5803.2 2) foot-pounds. 



(61.31 x 0.05 2?) - (14.7 x 0.15 2)) 



0.3 



] 



The actual engine will then give 0.9 x 5803.2 D = 5222.88 D foot-pounds per stroke 
or 5222.88 D x 100 foot-pounds per minute, which is to be made equal to 10 hp.,or 




C — 17.75 



Fig. 59. Art. 190. —Design of Air Engine. 



to 10 x 33,000 foot-pounds. Then 522,288 D = 330,000 and D = 0.63 cu. ft. Since 
the diameter of the engine equals its stroke, we write 0.7854 d 2 x d = 0.63 x 1728, 
where d is the diameter in inches; whence d = 11.15 in. 

To estimate the air consumption : at the point B, the whole volume of air is 
0.25 D. Part of this is clearance air, used repeatedly, and not chargeable to the 
engine. The clearance air at E had the volume V E and the pressure P E . If its 



96 



APPLIED THERMODYNAMICS 



behavior conforms to the law PV 1 - 8 — c, then at the pressure of 114.7 lb. (point G) 
we would have 1 

/ p \ L3 / 1 4. 7 \ 0.769 

P G V G « = P E V E « or V G = V B \j?) = 0.15 ^[~^) = 0.0309D. 

The volume of fresh air brought into the cylinder per stroke is then 

0.25 D - 0.0309 D = 0.2191 D 
or, per hour, 0.2191 x 0.63 x 100 x 60 = 828 cu. ft. Reduced to free air (Art. 186) f 

114 7 
this would be 828 x ' = 6450 cu. ft., or 645 cu. ft. per Ihp.-hr. (Compare 

Art. 192.) 14J 

191. Effect of Early Compression. If compression were to begin at a suffi- 
ciently early point, so that the pressure were raised to that in the supply pipe 
before the admission valve opened, the fresh air would find the clearance space 
already completely filled, and a less quantity of such fresh air, by 0.05 D, instead 
of 0.0309 D, would be required. 

192. Actual Performances of Air Engines. Kennedy (5) found a con- 
sumption of 890 cu. ft. of free air per Ihp.-hr., in a small horizontal steam 
engine. Under the conditions of Art. 183, the theoretical maximum work 
which this quantity of air could perform is 1.27 hp. The cylinder effi- 
ciency (Art. 215) of the engine was therefore 1.0-5-1.27=0.79. With 
small rotary engines, without expansion, tests of the Paris compressed air 
system showed free air consumption rates of from 1946 to 2330 cu. ft. 
By working these motors expansively, the rates were brought within 
the range from 848 to 1286 cu. ft. A good reciprocating engine with pre- 
heated air realized a rate of 477 cu. ft., corresponding to 36.4 lb., per 
brake horse power per hour. The cylinder efficiencies in these examples 
varied from 0.368 to 0.876, and the mechanical efficiencies (Art. 214) from 
0.85 to 0.92. 

The Air Compressor 

193. Action of Piston Compressor. Figure 60 represents the 
parts concerned in the cycle of an air compressor. Air is drawn 

from the atmosphere through the spring check 
valve a, filling the space O in the cylinder. This 
inflow of air continues until the piston has 
reached its extreme right-hand position. On the 
return stroke, the valve a being closed, compres- 
sion proceeds until the pressure is slightly greater 
than that in the receiver D. The balanced outlet 
valve b then opens, and air passes from O to I) 
at practically constant pressure. When the pis- 




Fig. 60. Art. 193.- 
Piston Compressor 



THE AIR COMPRESSOR 



97 



ton reaches the end of its stroke, there will still remain the clear- 
ance volume of air in the cylinder. This expands during the early 
part of the next stroke to the right, but as soon as the pressure of 
this air falls slightly below that of the atmosphere, the valve a again 
opens. p 

194. Cycle. An actual diagram is given, 
as AD OB, Fig. 61. Slight fluctuations in 
pressure occur during discharge along AD and 
during suction along CB; the mean discharge 
pressure must of course be slightly greater 
than the receiver pressure, and the mean suc- 
tion pressure slightly less than atmospheric pressure. Eliminating 
these irregularities and the effect of clearance, the ideal diagram 
is adcb. 




Fig. 61. Art. 104. — Cycle 
of Air Compressor. 



195. Form of Compression Curve. The remarks in Art. 180 as to 
the conditions of isothermal or adiabatic expansion apply equally to the 
compression curve BA. Close approximation to the isothermal path is the 

ideal of compressor per- 
formance. Let A, Fig. 62, 
be the poiut at which 
compression begins, and 
let DE represent the 
maximum pressure to be 
attained. Let the cycle 
be completed through the 
states F, G. Then the 
work expended, if com- 
pression is isothermal, is 
ACFG; if adiabatic, the 
work expended is ABFG. 
The same amount of air 




Fig. 62. 



Arts. 195, 197, 213, 218. — Forms of Compression 
Curve. 



has been compressed, and to the same pressure, in either case; the area 
ABC represents, therefore, needlessly expended work. Furthermore, dur- 
ing transmission to the point at which the air is to be applied, in the 
great majority of cases, the air will have been cooled down practically 
to the temperature of the atmosphere ; so that even if compressed adia- 
batically, with rise of temperature, to B, it will nevertheless be at the 
state C when ready for expansion in the consumer's engine. If it there 



98 APPLIED THERMODYNAMICS 

again expand adiabatically (along CH) instead of isothermally (along 
CA), a definite amount of available power will have been lost, repre- 
sented by the area CIIA. During compression, we aim to have the work 
area small; during expansion the object is that it be large; the adiabatic 
path prevents the attainment of either of these ideals. 

The loss of power by adiabatic compression is so great that various 
methods are employed to produce an approximately isothermal path. As 
a general rule, the path is consequently intermediate between the iso- 
thermal and the adiabatic, a poly tropic, pv n = C. The relations derived 
in Arts. 183 and 185 for adiabatic compression apply equally to this path, 
excepting that- for y we must write n, the value of n being somewhere 
between 1.0 and 1.402. The effect of water in the cylinder, whether in- 
troduced as vapor with the air, or purposely injected, is to somewhat 
reduce the value of n, to increase the interchange of heat with the walls, 
and to cause the line FG, Fig. 62, to be straight and vertical, rather than 
an adiabatic expansion, thus slightly increasing the capacity of the com- 
pressor, as shown in Art. 222. 

196. Temperature Rise. The rise of temperature due to compression may be 
computed as in Art. 185. Under ordinary conditions, the air leaves the com- 
pressor at a temperature higher than that of boiling water. Without cooling 
devices, it may leave at such a temperature as to make the pipes red hot. It is 
easy to compute the (not very extreme) conditions under which the rise in tem- 
perature would be sufficient to melt the cast-iron compressor cylinder. 

197. Computation of Loss. The uselessly expended work during adiabatic 
(and similarly, during any other than isothermal) compression may be directly 
computed from the difference of the work areas CAKI and CBAKT, Fig. 62. 
The work under the isothermal is (p, v, referring to the point C, and P, V, to 
the point ^4), pvlog e (V + v) = pvlog e (p -r- P) ; while if Q is the volume at B, 
the work under ABC is 

pQ-PV. 



p(Q - y) + - 



but 



P Q» = PV»3,nd Q= v(-Y; 



so that the percentage of loss corresponding to any ratio of initial and final pres- 
sures and any terminal (or initial) volume may be at once computed. 

198. Basis of Methods for Improvement. Any value of n exceeding 1.0 for 
the path of compression is due to the generation of heat as the pressure rises, 
faster than the walls of the cylinder can transmit it to the atmosphere. The high 
temperatures thus produced introduce serious difficulties in lubrication. Economi- 
cal compression is a matter of air cooling ; while, on the consumer's part, economy 
depends upon air healing. 



COMPRESSION CURVE 



99 



199. Air Cooling. In certain applications, where a strong draft is available, 
the movement of the atmosphere may be utilized to cool the compressor cylinder 
walls and thus to chill the working air during compression. While this method 
of cooling is quite inadequate, it has the advantage of simplicity and is largely 
employed on the air "pumps" which operate the brakes of railway trains. 



200. Injection of Water. This was the method of cooling originally em- 
ployed at Mont Cenis by Colladon. Figure 63 shows the actual indicator card 
(Art. 484) from one of the older Colladon ,p 
compressors. EB CD is the corresponding- 
ideal card with isothermal compression. 
The cooling by stream injection was evi- 
dently not very effective. Figure 64 rep- 
resents another diagram from a compressor 
in which this method of cooling was em- 
ployed ; ab representing the isothermal and 
ac the adiabatic. The exponent of the 
actual curve ad was 1.36; the gain over 
adiabatic compression was very slight. By 
p 




Fig. 63. 




Art. 200. — Card from Colladon 
Compressor. 



Fig. 64. 



Art. 200. — Cooling by 
Injection. 



introducing the 
water in a very 

fine spray, a somewhat lower value of the exponent 
was obtained in the compressors used by Colladon on 
the St. Gothard tunnel. Gause and Post (6) have re- 
duced the value of n to 1.26 by an atomized spray. 
Figure 65 shows one of their diagrams, ab being the 
isothermal and ac the adiabatic. In all cases, 
spray injection is better than solid stream in- 
jection. The value n = 1.36, above given, 
was obtained when a solid jet of half -inch 
diameter was used. It is estimated that errors 
of the indicator may introduce an uncer- 
n. Piston leakage would cause an 
p 



Jet 



tainty amounting to 0.02 in the value of 
apparently low value. The comparative 
efficiency of spray injection is shown from 
the fairly uniform temperature of dis- 
charged air, which can be maintained even 
with a varying speed of the compressor. 
In the Gause and Post experiments, with 
inlet air at 81|° F., the temperature of dis- 
charge was 95° F. Spray injection has the 
objection that it fills the air with vapor, and 
it has been found that the orifices must be 
so small that they soon clog and become 
inoperative. The use of either a spray or 
a solid jet causes cutting of the cylinder and piston by the gritty substances carried 
in the water. In American practice the injection of water has been abandoned. 




Fig. 65. Art. 200. — Cooling by Atomized 
Spray. 



100 



APPLIED THERMODYNAMICS 




201. Water Jackets. These reduce the value of n to a very slight ex- 
tent only, but are generally employed because of their favorable influence 

on cylinder lubrication. Gause and 
Post found that with inlet air at 
81° F., and jackets on the barrels of 
the cylinders only (not on the heads), 
the temperature of the discharged air 
was 320° P. Cooling occurred dur- 
ing expulsion rather than during com- 
pression. The cooling effect depends 
largely upon the heat transmissive 
power of the cylinder walls, and the 

value of n consequently increases at 
Fig. 66. Art. 201. — Cooling by Jackets. , . , j m j 

high speeds. Two specimen cards 

are given in Pig. 66 ; ab being the isothermal and ac the adiabatic. With 
more thorough cooling, jacketed 
heads, etc., a lower value of n 
may be obtained ; but this value 
is seldom or never below 1.3. 
Figure 67 shows a card given 
by Unwin from a Cockerill com- 
pressor, DC indicating the ideal 
isothermal curve. At the 
higher pressures, air is appar- 
ently more readily cooled; its 
own heat-conducting power is 
increased. 

202. Heat Abstracted. In ._JS!H«2^«««!™_ 
Fig. 68, let AB and AC be the Fig. 67. Art. 201. — Cockerill Compressor with 
adiabatic and the actual paths, Jacket Cooling - 

An and CN adiabatics ; the heat to be abstracted is then equivalent to 





NCAn = IACL + nAIE 



Now IACL=^- 



PV 



, nAIE = 



NCLE. 
PV 



y 



NCLE = -£® 



^-1' 

pv 



L I E 

Fig. 68. Arts. 202, 203.— Heat Ab- 
stracted by Cooling Agent. 



whence 

n — 1 y — 1 ?/-— 1 

This is the heat to be abstracted per 
volume Tat pressure P, compressed to 



MULTI-STAGE COMPRESSION 



101 



p, expressed in foot-pounds. For isothermal compression, as along 
AD, IACL=pv log e (V-i-v), and the total heat to be abstracted is measured 
by this area. If the path is adiabatic, AB, n = y, and the expression for 
heat abstraction becomes zero. # 

203. Elimination of v. It is convenient to express the total area NCAn in 
terms of p, P, and V only. The area 

n _l n-l(pv-PV) n-l\PV ' n-\\\p) l \ 

Also, 

NCLE = J^-=*^[P% 
y-\ y-l\pj 

whence NCAn = 22 \ (2)*f - {] + 22. _ J>2(L)L 
n—lL\P/ J y — 1 y—\\p) 

204. Water Required. Let the heat to be abstracted, as above com- 
puted, be H, in heat units. Then if S and s are the final and initial 
temperatures of cooling water, and C the weight of water circulated, we 
have C= Hs-(S — s), the specific heat of water being taken as 1.0. In 
practice, the range of temperature of the cooling water may be from 40° 
to 70° F. 

205. Multi-stage Compression. The effective method of securing a 
low value of n is by multi-stage operation, the principle of which is 
illustrated in Fig. 69. Let A be the 
state at the beginning of compres- 
sion, and let it be assumed that the 
path is practically adiabatic, in spite 
of jacket cooling, as AB. Let AC 
be an isothermal. In multi-stage 
compression, the air follows the path 
AB up to a moderate pressure, as at 
D, and is then discharged and cooled 
at constant pressure in an external 
vessel, until its temperature is as 
nearly as possible that at which it was admitted to the cylinder. 
The path representing this cooling is DE. The air now passes to 



CH 




Fig. 



Art. 205. — Multi-stage Com- 
pression. 



* More simply, as suggested by Chevalier, the specific heat along AC is s = I 



(Art. 112): the heat to be abstracted is then 



s(T-t) 



i O - y) 

B(n-\j 



(PV-pv). 



102 



APPLIED THERMODYNAMICS 




Fig. 70. Arts. 205, 206.— Two-stage Com- 
pressor Indicator Diagram. 



a second cylinder, is adiabatically compressed along FJF, ejected and 
cooled along FGr, and finally compressed in still another cylinder 

along GrH. The diagram illus- 
trates compression in three 
" stages " ; but two or four stages 
are sometimes used. The work 
saved over that of single stage 
adiabatic compression is shown 
by the irregular shaded area 
HGrFFDB, equivalent to a re- 
duction in the value of w, under 
good conditions, from 1.402 to 
about 1.25. Figure 70 shows the diagram from a two-stage 2000 hp. 
compressor, in which solid water jets were used in the cylinders. 
The cooling water was at a lower 
temperature than the inlet air, 
causing the point h to fall inside 
the isothermal curve AB. The 
compression curves in each cyl- 
inder give w = 1.36. Figure 71 
is the diagram for a two-stage 
Riedler compressor with spray in- 
jection, AB being an isothermal 

Fig. 71. Arts. 205, 214. — Two-stage Riedler 
and AG an adiabatlC. Compressor Diagram. 




206. Intercooling. Some work is always wasted on account of the friction of 
the air passing through the intercooling device. In early compressors, this loss 
often more than outweighed the gain due to compounding. The area ghij, Fig. 
70, indicates the work wasted from this cause. In this particular instance, the 
loss is exceptionally small. Besides this, the additional air friction through two 
or more sets of valves and ports, and the extra mechanical friction due to a multi- 
plication of cylinders and reciprocating parts must be considered. Multi-stage 
compression does not pay unless the intercooling is thoroughly effective. It seldom 
pays when the pressure attained is low. Incidental advantages in multi-stage 
operation arise from reduced mechanical strains (Art. 462), higher volumetric 
efficiency (Art. 226), better lubrication, and the removal of moisture by precipita- 
tion during the intercooling. 



207. Types of Intercoolers. The "external vessel" of Art. 205 is called the 
intercooler. It consists usually of a riveted or cast-iron cylindrical shell, with cast- 



INTERCOOLING 



103 



iron heads. Inside are straight tubes of brass or wrought iron, running between 
steel tube sheets. The back tube sheet is often attached to a stiff cast-iron inter- 




Fig. 72. Art. 207. — Allis-Chalmers Horizontal Intercooler. 

nal head, so that the tubes, sheet, and head 
are free to move as the tubes expand 
(Fig. 72). The air entering the shell sur- 
rounds the tubes and is compelled by baffles 
to cross the tube space on its way to the out- 
let. Any moisture precipitated is drained 
off at the pipe a. The water is guided to 
the tubes by internally projecting ribs on 
the heads, which cause it to circulate from 
end to end of the intercooler, several times. 
If of ample volume, as it should be, the 
intercooler serves as a receiver or storage 
tank. The one illustrated is mounted in 
a horizontal position. A vertical type is 
shown in Fig. 73. The funnel provides a 
method of ascertaining at all times whether 
water is flowing. 

RIR OUTLET *=s 

208. Aftercoolers. In most 
manufacturing plants, the pres- 
ence of moisture in the air is ob- 
jectionable, on account of the 
difficulty of lubrication of air 
tools, and because of the rapid de- 
struction of the rubber hose used 
for connecting these tools with 
the pipe line. To remove the 
moisture (and some of the oil) FlG . 73< Art. 207.- Ingersoll-Sergeaut Vertical 
present after the last stage of com- Intercooler. 




104 APPLIED THERMODYNAMICS 

pression, and by cooling the air to decrease the necessary size of transmitting pipe, 
aftercoolers are employed. They are similar in design and appearance to inter- 
coolers. An incidental advantage arising from their use is the decreased strain 
on the pipe line following the introduction of air at a more nearly normal tem- 
perature. 

209. Power Consumed. From Art. 98, the work under any curve 

2 w n =PV n is, adopting the notation of Art. 202, P"- p] f 

n — 1 

= pv A PV\ pv h ^V""}. 
n — 1 \ pv J n. — l\ \pj ] 

The work along an adiabatic is expressed by the last formula if we make 
n = y = 1.402. The work of expelling the air from the cylinder after com- 
pression is pv. The work of drawing the air into the cylinder, neglecting 

n-l 

clearance, is PV—pvl — ) • The net work expended in the cycle is the 

vw 

algebraic sum of these three quantities, which we may write, 

n.-ll W J \PJ »-l I \pj J 

It is usually more convenient to eliminate v, the volume after compres- 
sion. This gives the work expression, 

n-l 

PVn [ fp\ n _ 1 



n-l {\P t 

If pressures are in pounds per square inch, the foot-pounds of work per 
minute will be obtained by ninl tiphying this expression by the number of 
working strokes per minute and by 144 ; and the theoretical horse power 
necessary for compression may be found by dividing this product by 
33,000. If Ave make V=l, P=14.7, we obtain the power necessary to 
compress one cubic foot of free air. If the air is to be used to drive a 
motor, it will then in most cases have cooled to its initial temperature 
(Art. 195), so that its volume after compression and cooling will be 
PV-7-p. The work expended per cubic foot of this compressed and 
cooled air is then obtained by multiplying the work per cubic foot of free 

air by ±- • 
J P 

210. Work of Compression. In some text-books, the work area under the 
compression curve is specifically referred to as the work of compression. This is 
not the total work area of the cycle. 



RECEIVER PRESSURE 105 

211. Range of Stages in Multi-stage Compression. Let the lowest pres- 
sure be q, the highest p, and the pressure during intercooling P. Also let 
intercooling be complete, so that the air is reduced to its initial tempera- 
ture, so that the volume V after intercooling is ^, in which r is the 
volume at the beginning of compression in the first cylinder. Adopting 

the si 
have 



the second of the work expressions just found, and writing z for , we 



Work in first stage = ^- \ ( — ) — 1 I 



-5 ST 



Work in second stage = U ^) — ll=— -if— ]— ll 

« [\PJ J z [\PJ j 



Total .o rk = ? {(|y + (|)'-2 

Differentiating with respect to P, we obtain 

dW 
dP 



= W. 



_ qr 

z 


X5Tf-*r*] 


-mr-mi 


= qr 


' p*- 1 p* 1 m 
qz pz+ l j • 



For a minimum value of W, the result desired in proportioning the pres- 
sure ranges, this expression is put equal to zero, giving 

P 2 =2iq, or P = ^/pq. 

An extension of the analysis serves to establish a division of pressures 
for four stage machines. From the pressure ranges given, it may easily 
be shown that in the ideal cycle the condition of minimum work is that the 
amounts of work done in each of the cylinders be equal. The number of 
stages increases as the range of pressures increases; in ordinary practice, 
the two-stage compressor is employed, with final pressures of about 100 
lb. per square inch above the pressure of the atmosphere. 

Engine and Compressok Relations 

212. Losses in Compressed Air Systems. Starting with mechanical power 
delivered to the compressor, we have the following losses : — 
(a) friction of the compressor mechanism, affecting the mechanical 
efficiency; 



106 APPLIED THERMODYNAMICS 

(b) thermodynamic loss, chiefly from failure to realize isothermal com- 

pression, but also from friction and leakage of air, clearance, etc., 
indicated by the cylinder efficiency; 

(c) transmissive losses in pipe lines ; 

(d) thermodynamic losses at the consumer's engine, like those of (b) ; 

(e) friction losses at the consumer's engine, like those of (a). 

213. Compressive Efficiency. While not an efficiency in the true sense of the 
term, the relation of work generated during expansion in the engine to that ex- 
pended during compression in the compressor is sometimes called the compressive 
efficiency. It is the quotient of the areas FCHG and FBAG, Fig. 62. From the 
expression in Art. 209 for work under a polytropic plus work of discharge along 
BF or of admission along FC, we note that, the values of P and p being identical 
for the two paths, AB and CH, in question, the total work under either of these 
paths is a direct function of the volume V at the lower pressure P. In this case, 
providing the value of n be the same for both paths, the two work areas have the 
ratio V 7- x, where V is the volume at A , and x that at H. It follows that all the 
ratios of volumes LN -4- LM, OQ -=- OP, etc., are equal, and equal to the ratio of 

areas. The compressive efficiency, then, = — = T -4- t, where t is the temperature 

at A (or that at C), and T that at H. For isothermal paths, T = t, and the com- 
pressive efficiency is unity. In various tests, the compressive efficiency has ranged 
from 0.488 to 0.898. It depends, of course, on the value of n ; increasing as n de 
creases. 

214. Mechanical Efficiency. For the compressor, this is the quotient of work 
expended in the cylinder by work consumed at the fly wheel ; for the engine, it 
is the quotient of work delivered at the fly wheel by work done in the cylinder. 

Friction losses in the mechanism measure the mechanical inefficiency of the 
compressor or engine. With no friction, all of the power delivered would be ex- 
pended in compression, and all of the elastic force of the air would be available 
for doing work, and the mechanical efficiency would be 1.0. In practice, since 
compressors are usually directly driven from steam engines, w T ith piston rods in 
common, it is impossible to distinguish between the mechanical efficiency of the 
compressor and that of the steam engine. The combined efficiency, in one of the 
best recorded tests, is given as 0.92. For the compressor w 7 hose card is shown in 
Fig. 71, the combined efficiency was 0.87. Kennedy reports an average figure of 
0.845 (7). U 11 win states that the usual value is from 0.85 to 0.87 (8). These 
efficiencies are of course determined by comparing the areas of the steam and air 
indicator cards. 

215. Cylinder Efficiency. The true efficiency, thermodynamically speaking, 
is indicated by the ratio of areas of the actual and ideal PV diagrams. For the 
compressor, the cylinder efficiency is the ratio of the work done in the ideal cycle, 
without clearance, drawing in air at atmospheric pressure, compressing it isothermally 
and discharging it at the constant receiver pressure, to the work done in the actual cycle 
of the same maximum volume. It measures item (b) (Art. 212). It is not the "com- 



PLANT EFFICIENCY 107 

pressive efficiency " of Art. 213. For the engine, it is the ratio of the work done in 
the actual cycle to the work of an ideal cycle without clearance, with isothermal expan- 
sion to the same maximum volume from the same initial state, and with constant pressures 
during reception and discharge ; the former being that of the pipe line and the latter that 
of the atmosphere. Its value may range from 0.70 to 0.90 in good machines, in gen- 
eral increasing as the value of n decreases. An additional influence is fluid fric- 
tion, causing, in the compressor, a fall of pressure through the suction stroke and 
a rise of pressure during the expulsion stroke .; and in the engine, a fall of pressure 
during admission and excessive back pressure during exhaust. All of these condi- 
tions alter the area of the PV cycle. In well-designed machines, these losses 
should be small. A large capacity loss in the cylinder is still to be considered. 

216. Discussion of Efficiencies. Considering the various items of loss sug- 
gested in Art. 212, we find as average values under good conditions, 

Qa) mechanical efficiency, 0.85 to 0.90; say 0.85; 

(&) cylinder efficiency of compressor, 0.70 to 0.90; say 0.80; 

(c) transmission losses, as yet undetermined ; 

(cT) cylinder efficiency of air engine, 0.70 to 90.0; say 0.70; 

(e) mechanical efficiency of engine, 0.80 to 0.90; say 0.80. 

The combined efficiency from steam cylinder to work performed at the con- 
sumer's engine, assuming no loss by transmission, would then be, as an average, 

0.85 x 0.80 x 0.70 x 0.80 = 0.3808. 

For the Paris transmission system, Kennedy found the over-all efficiency (includ- 
ing pipe line losses, 4 per cent) to be 0.26 with cold air or 0.384 with preheated 
air, allowing for the fuel consumption in the preheaters (9). 

217. Maximum Efficiency. In the processes described, the ideal efficiency in 
each case is unity. We are here dealing not with thermodynamic transformations 
between heat and mechanical energy, but only with transformations from one form 
of mechanical energy to another, in part influenced by heat agencies. No strictly 
thermodynamic transformation can have an efficiency of unity, ou account of the 
limitation of the second law. 

218. Entropy Diagram. Figure 62 may serve to represent the com- 
bined ideal PV diagrams of the compressor (GABF) and engine (FCHG). 

The quotient is the compressive efficiency. The area representing 

net expenditure of work is CBAH, bounded ideally by two adiabatics or in 
practice by two poly tropics (not ordinarily isodiabatics) and two paths of 
constant pressure. This area is now to be illustrated on the TN coordi- 
nates. 



108 



APPLIED THERMODYNAMICS 




For convenience, we reproduce the essential features of Fig. 62 
in Fig. 74. In Fig. 75, lay off the isothermal T, and choose the 

point A at random. Now 
if either T B or T H be 
given, we may complete 
the diagram. Assume 
that the former is given ; 
then plot the correspond- 
ing isothermal in Fig. 75. 
Draw AB, an adiabatic, 
BO and AH as lines 
cf constant pressure 

Fig. 74. Art. 218. — Engine and Compressor Diagrams. ln=k log e — J, the point 

falling on the isothermal T. Then draw OH, an adiabatic, de- 

T T 

termining the point H; or, from Art. 213, noting that — - = -= d , we 

T c T 

may find the point H di- 
rectly. If the paths AB 
and OH are not adia- 
batics, we may compute 
the value of the specific 
heat from that of n and 
plot these paths on Fig. 
75 as logarithmic curves ; 
but if the values of n are 
different for the two 
paths, it no longer holds 

The area 




that |# = ^ 



T f 



Fig. 75. Arts. 218, 219, 221.— Compressed Air System, 
Entropy Diagram. 



OBAH in Fig. 75 now represents the net work expenditure in 
heat units. 



219. Comments. As the exponents of the paths AB and CH decrease, 
these paths swerve into new positions, as AE, CD, decreasing the area 
representing work expenditure. Finally, with n = l, isothermal paths, 
the area of the diagram becomes zero ; a straight line, CA. Theoretically, 
with water colder than the air, it might be possible to reduce the tempera- 



ENTROPY DIAGRAMS 



109 



ture of the air during compression, giving such a cycle as AICDA, or even, 
with isothermal expansion in the engine, AICA; in either case, the net 
work expenditure might be nega- 
tive; the cooling water accomplish- 
ing the result desired. 

220. Actual Conditions. Under 
the more usual condition that the 
temperature of the air at admission 
to the engine is somewhat higher 
than that at which it is received by 
the compressor, we obtain Figs. 
76, 77. T, T c and either T B or T H 
must now be given. The cycle in 
which the temperature is reduced 
during compression now appears FlG . 76 . Art . 220. -Usual Combination of 
as AICDA or AHA. Diagrams. 

T B 





Fig. 



Art. 220. — Combined Entropy Diagrams. 



221. Multi-stage Operation. Let the ideal pv path be DECBA, Fig. 78. 
The "triangle" ABC of Fig. 75 is then replaced by the area DECBA, 
Fig. 79, bounded by lines of constant pressure and adiabatics. The area 








1 










j/t 








E 


' | 


E 




A- — 




^/k 


D 


- 




( 









Fig. 78. Art. 221. — Three-stage Com- 
pression and Expansion. 



Fig. 79. Art. 221 . — Entropy Diagram , 
Multi-stage Compression. 



110 



APPLIED THERMODYNAMICS 



saved is BFEC, which approaches zero as the pressure along CE, Fig. 78, 
approaches that along AB or at D, and becomes a maximum at an inter- 
mediate position, already determined in 
Art. 211. With inadequate intercooling, 
the area representing work saved would be 
yFEx. Figures 80 and 81 represent the 
ideal pv and nt diagrams respectively for 
compressor and engine, each three-stage, 
with perfect intercooling and aftercooling 
and preheating and with no drop of pres- 
sure in transmission. BbA and AhB 
would be the diagrams with single-stage 
adiabatic compression and expansion. 




Fig. 80. Art. 221. — Three-stage 
Compression and Expansion. 



T 














A 


r 


t/^ 
l^^ 


e 


c 
B 










9 


» 


i 


, 



Fig. 81. Art. 221. — Three-stage Compression and Expansion. 

Compressor Capacity 

222. Effect of Clearance on Capacity. Let A BCD, Fig. 57, be the ideal pv dia- 
gram of a compressor without clearance. If there is clearance, the diagram will 
be aBCE ; the air left in the cylinder at a will expand, nearly adiabatically, along 
aE, so that its volume at the intake pressure will be somewhat like BE. The 
total volume of fresh air taken into the cylinder cannot be DC as if there were no 
clearance, but is only EC. The ratio EC '.DC is called the volumetric efficiency. 
It is the ratio of free air drawn in to piston displacement. 

223. Volumetric Efficiency. This term is sometimes incorrectly applied to the 
factor 1 — c, in which c is the clearance, expressed as a fraction of the cylinder 
volume. This is illogical, because this fraction measures the ratio of clearance air 
at final pressure, to inlet air at atmospheric pressure (A a — DC, Fig. 57) ; while 
the reduction of compressor capacity is determined by the volume of clearance air 
at .atmospheric pressure. If the clearance is 3 per cent, the volumetric efficiency is 
much less than 97 per cent. 



224. Friction and Compressor Capacity. If the intake ports or pipes are small, 
an excessive suction will be necessary to draw in the charge, and the cylinder will 



VOLUMETRIC EFFICIENCY 



111 



be filled with air at less than atmospheric pressure. Its equivalent volume at 
atmospheric pressure will then be less than that of the cylinder. This is shown 
in Fig. 82. The line of atmospheric pressure is DF, the capacity is 
reduced by FG, and the volumetric efficiency is DF -^ HG. The capacity 
may be seriously affected from this cause, in the case of a badly designed 
machine. 



225. Volumetric Efficiency ; Other Factors. 




Fig. 82. Art. 224. — Effect of Suction Friction. 



Where jackets or water jets 
are used, the air is often 
somewhat heated during 
the intake stroke, increas- 
ing its volume, and thus, 
as in Art. 224, lowering 
the volumetric efficiency. 
The effect is more notice- 
able with jacket cooling, 
with which the cylinder 
walls often remain con- 
stantly at a temperature above that of boiling water. Tests have shown a loss 
of capacity of 5 per cent, due to changing from spray injection to jacketing. — A 
high altitude for the compressor results in its being supplied with rarefied air, and 
this decreases the volumetric efficiency as based on air under standard pressure. 
At an elevation of 10,000 ft. the capacity falls off 30 per cent. This is sometimes 
a matter of importance in mining applications. — Volumetric efficiency, in good 
designs, is principally a matter of low clearance. The clearance of a cylinder is 
practically constant, regardless of its length ; so that its percentage is less in the 
case of the longer stroke compressors. Such compressors are comparatively 
expensive. — When water is injected into the cylinder, as is often the case in 
European practice, the clearance space may be practically filled with water at the 
end of the discharge stroke. Water does not appreciably expand as the pressure 
is lowered ; so that in these cases the volumetric efficiency may be determined by 
the expression 1 — c of Art. 223, being much greater than in cases where water 
injection is not practiced. 



226. Volumetric Efficiency in Multi-stage Compression. Since the effect 
of multi-stage compression is to reduce the pressure range, the expansion 
of the air caught in the clearance space is less, and the distance DE, 
Fig. 57, is reduced. This makes the volumetric efficiency, EC -h DC, 
greater than in single stage cylinders. If FGH represent the line of in- 
termediate pressure, the ratio JE -r- DCis the gain in volumetric efficiency. 



227. Refrigeration of Entering Air. Many of the advantages following multi- 
stage operation and intercooling have been otherwise successfully realized by the 
plan of cooling the air drawn into the compressor. This of course increases the 
density of the air at atmospheric pressure, and greatly increases the volumetric 
efficiency. Incidentally, much of the moisture is precipitated. At the Isabella 



112 



APPLIED THERMODYNAMICS 



furnace of the Carnegie Steel Company, at Etna, Pennsylvania, a plant of this 
kind has been installed. An ordinary ammonia refrigerating machine cools the 
air from 80° to 28° F. This should decrease the specific volume in the ratio 
(459.6 + 28) -r- (459.6 + 80) = 0.90. The free air capacity should consequently 
be increased in about this ratio (10). 

228. Typical Values. Excluding the effect of clearance, a loss in ca- 
pacity of from 6 to 22 per cent has been found by Unwin (11) to be due 
to air friction losses and to heating of the entering air. Heilemann (12) 
finds volumetric efficiencies from 0.73 to 0.919. The volumetric efficiency 
could be precisely determined only by measuring the air drawn in and 
discharged. 

229. Volumetric and Thermodynamic Efficiencies. The volumetric effi- 
ciency is a measure of the capacity only. It is not an efficiency in the sense 
of a ratio of " effect " to " cause." In Pig. 83 the solid lines show an actual 
compressor diagram, the dotted lines, EGHB, the corresponding perfect 
diagram, with clearance and isothermal compression. In the actual case 
we have the wasted work areas, 

HLJQ, due to friction in discharge ports ; 
GQKD, due to non-isothermal compression; 
DFMC, due to friction during the suction of the air. 

At BHC, there is an area representing, apparently, a saving in work 
expenditure, due to the expansion of the clearance air ; this saving in 

work has been accomplished, however, 
with a decreased capacity in the pro- 
portion BC -5- BE, a proportion which 
is greater than that of BHC to the total 
work area. Further, expansion of the 
clearance air is made possible as a result 
of its previous compression along FDK; 
and the energy given up by expansion 
can never quite equal that expended in 
compression. The effect of excessive 
friction during suction, reducing the 
capacity in the ratio DE -s- BE, is 
usually more marked on the capacity than on the work. Both suction 
friction and clearance decrease the cylinder efficiency as well as the 
volumetric efficiency, but the former cannot be expressed in terms of 
the latter. In fact, a low volumetric efficiency may decrease the work 
expenditure absolutely, though not relatively. An instance of this is found 
in the case of a compressor working at high altitude. Friction during dis- 




FlG 



83. Art. 229. — Volumetric and 
Thermodynamic Efficiencies. 



COMPRESSOR DESIGN 113 

charge decreases the cylinder efficiency (note the wasted work area HLJQ), 
but is practically without effect on the capacity. 

Compressor Design 

230. Capacity. The necessary size of cylinder is calculated much as in 
Art. 190. Let p, v, t, be the pressure, volume, and temperature of dis- 
charged air (y meaning the volume of air handled per minute), and P, F, T, 
those of the inlet air. Then, since PV-r- T = pv -f- t, the volume drawn 
into the compressor per minute is V ' = pvT ■+- Pt, provided that the air is 
dry at both intake and delivery. If n is the number of revolutions per 
minute, and the compressor is double-acting, then, neglecting clearance, 

vvT 
the piston displacement per stroke is V-z- 2 n = +- . 

This computation of capacity takes no account of volumetric losses. 
In some cases, a rough approximation is made, as described, and by 
slightly varying the speed of the compressor its capacity is made equal to 
that required. Allowance for clearance may readily be made. Let the 
suction pressure be P, the final pressure p, the clearance volume at the 

final pressure — of the piston displacement. Then, if expansion in the 
m 

clearance space follows the law pv n = PV n , the volume of clearance air 

at atmospheric pressure is 






mJ\P. 

of the piston displacement. For the displacement above given, we there- 
fore write, 

i AVA r 



v_ 

2n 



1 + 



This may be increased 5 to 10 per cent, to allow for air friction, air 
heating, etc. 

231. Design of Compressor. The following data must be assumed : 

(a) capacity, or piston displacement, 

(6) maximum pressure, 

(c) initial pressure and temperature, 

(d) temperature of cooling water, 

(e) gas to be compressed, if other than air. 

Let the compressor deliver 300 cu. ft. of compressed air, measured 
at 70° F., per minute, against 100 lb. gauge pressure, drawing its supply at 
14.7 lb. and 70° F., the clearance being 2 per cent. Then, ideally, the free 



114 



APPLIED THERMODYNAMICS 



air per minute will be 300 x (114.7 -=- 14.7) = 2341 cu. ft., or allowing 5 
per cent for losses due to air friction and heating during the suction, 
2341 -=- 0.95 = 2464 cu. ft. To allow for clearance, we may use the ex- 
pression in Art. 230, making the displacement, with adiabatic expansion 
of the clearance air, 

2464 -*- [1-0.02 f^l\k + 0.02] = 2640 cu. ft. 

Assuming for a compressor of this capacity a speed of 80 r. p. m., the 
necessary piston displacement for a double-acting compressor is then 
2640 -i-(2x 80) = 16.5 cu. ft. per stroke, or for a stroke of 3 ft., the piston 
area would be 792 sq. in. (13). The power expended for any assumed 
compressive path may be calculated as in Art. 190, and if the mechanical 
efficiency be assumed, the power necessary to drive the compressor at 
once follows. The assumption of clearance as 2 per cent must be justified 
in the details of the design. The elevation in temperature of the air may 
be calculated as in Art. 185, and the necessary amount of cooling water 
as in Art. 203, the exponents of the curves being assumed. 

232. Two-stage Compressor. From Art. 211 we may establish an inter- 
mediate pressure stage. This leads to a new correction for clearance, and 
to a smaller loss of capacity due to air heating. Using these new values, 
we calculate the size of the first-stage cylinder. For the second stage, the 
maximum volume may be calculated on the basis that intercooling is com- 
plete, whence the cylinder volumes are inversely proportional to the suc- 
tion pressures. The clearance correction will be found to be the same as 
in the low-pressure cylinder. The capacity, temperature rise, water con- 
sumption, power consumption, etc., are computed as before. A considera- 
ble saving in power follows the change to two stages. 



233. Problem. Find the cylinder dimensions and power consumption of a 
double-acting single-stage air compressor to deliver 4000 cu. ft. of free air per 

minute at 100 lb. gauge pres- 
sure at 80 r. p. m., the intake 
air being at 13.7 lb. absolute 
pressure, the piston speed 
640 ft. per minute, clearance 
4 per cent, and the clearance- 
expansion and compression 
curves following the law 

Lay off the distance GH, 
Fig. 84, to represent the (un- 
known) displacement of the 
Art. 233.— Design of Compressor. piston, which we will call D. 




COMPRESSOR DESIGN 



115 



Since the clearance is 4 per cent, lay off GZ = 0.04 2), determining ZP as a 
coordinate axis. Draw the lines TU, VW, YX, representing the absolute pres- 
sures indicated. The compression curve CE may now be drawn through C, and 
the clearance expansion curve DI through D. The ideal indicator diagram is 
CEDI. We have, 



P D V D 1 ' S5 =P I V I 1 ' 35 or V T 



$r v -(wr»> 



04 D= 0.1927 2). 



/ P~\°'" 4 / 13 7 \ 0.74 

P m VJ»=P V<?m or V E = (j^J 7* = (j^J 1.042) = 0.21582). 



.74 



,^«=P J> F^-«orF^=^j r*>={jpfY 0.04 2) -0.1829 2). 
PmVj*=P a Vf'» or F* = ^ W V a =(jH) 0,M i.042>= ; 0.i 



•2 2). 



But ^4i?= Fg — Fa = 0.8043 2) is the volume of free air drawn into the cylinder: 
AB-t-GH =0.804:3 is the volumetric efficiency : to compress 4000 cu. ft. of free air per 
minute the piston displacement must then be 4000 -=- 0.8043 = 4-973 cu. ft. per minute. 
Since the compressor is double-acting, the necessary cylinder area is the quotient 
of displacement by piston speed or 4973 -f- 640, giving 7.77 sq. ft., or (neglecting 
the loss of area due to the piston rod), the cylinder diameter is 37.60 in. From the 
conditions of the problem, the stroke is 640 -f(2x 80) = 4 ft. 

For the power consumption, we have 

W = GDEF+ FECH - JICH - GDIJ 

P E V E - P C Vc j> rl r rrs P D V D - PjVj 



= P E (V E - V D ) + 



0.35 



= 144 2) [(114.7x0.1758) + 



(114.7 x 0.2158) 



0.35 
(13.7 x 1.04) 



-(13.7 x 0.8473)- 
= 144 D [20.16 + 30.01 - 11.61 - 5.59] = 144 D x 32.97. 

This is the work for a piston displacement 
per minute, the horse power 
consumed in compression is 
144 x 32.97 x 4973 



0.35 

(114.7 x 0.04) -(13.7 x 0.1927) 



0.35 



2) cubic feet. Tf we take 2) at 4973 



33000 



= 715. 



234. Design of a Two- 
stage Machine. With condi- 
tions as in the preceding, con- 
sider a two-stage compressor 
with complete intercooling and 
a uniform friction of one pound 
between the stages. Here the 
combined diagrams appear as 
in Fig. 85. For economy of 
power, the intermediate pres- 



p 

E 




c 








1° 
1JF 




\° 

\k g 








I 






\ \ 






' 


V 




""Hi 






\ 








Q 






H 






0.04 






^^ 








D 













Fig. 85. Art. 234. — Design of Two-stage Compressor. 



116 APPLIED THERMODYNAMICS 



sure is Vlll.7 x 13.7 = 39.64, whence the first-stage discharge pressure and the 
second-stage suction pressure, corrected for friction, are respectively 40.14 and 
39.14 lb. For the first stage, Fig. 85, 

P F = P G = 40.14, P A = P B = 14.7, P q = P H = 13.7, V H = 1.04 A V F = 0.04 D. 

I p \ 0.74 / 1Q7 \ 0.74 _ 

P G V G ^ = P H V H ^ or V G = (i/j V s = l^^j 1.04 D = 0.4701 2). 

/p\0.74 /40 14\0.74 _ 

P q V q ^ = P F V F ^ or V q = f^j V F = (™±Z\ 0.04 D = 0.08864 D. 

/p \0.74 /4.O14\0.74 

Pa JY- 35 = Pi.IV- 35 or V A = ly\ V F = Hjjy ) 0.04 D = 0.08412 2). 

P B V B ™» = P H V B ^ or V B = (^r)°' 74 T 7 ir = (fH) 1 - 04 D = 0.9872). 

The volumetric efficiency is AB - D = (V B - V A ) -2) = 0.987 - 0.08412 = 0.90288. 
The piston displacement per minute is 4900 -f- 0.903 = 4430. The piston diameter 
is V(4430 + 640) x 144 -=- 0.7854 = 35.6 in. for a stroke of 630 v(2x 80) = 4 ft- 
The power consumption for this first stage is, 

W = P G (V G - V f ) + PgVg ~ P * Vb - P H (V S - V Q )-^ F 



n-1 
= [40.14(0.4701 - 0.04) + (40.14x0.4701W13.7xl.04) 

- 13.7(1.04 - 0.0886) - (40.14 x 0.04) -(13.7 x 0.0886)j 1M D 

= 2348.64 D foot-pounds or 10,404,475 foot-pounds per minute, equivalent to 
315.3 horse power. 

Second Stage 

Complete intercooling means that at the beginning of compression in the sec- 
ond stage the temperature of the air will be as in the first stage, 70° F. The 

P 13 7 

volume at this point will then be V z = ^ V n = -±^- 1.04 D = 0.364 D. We thus 
F P z 39.14 

locate the point Z, Fig. 85, and complete the diagram ZCEI, making V E — 0.04 

( y z - V E ) =0.01456 D, P C =P E = 114.7, P I= P Z = 39.14, and compute as follows : 



'^)°' 74 0.3642) -0.3574 2). 



PzV'^y _/3_9.14\ - 74 , 
P K ) ~ V40. 



/p \0.74 /3Q14\0.74 _ _ 

P C IV' 35 = P Z V Z ^ or V c = [^ J V z = [f^j 0.364 2) = 0.1642 2). 

ip \0.74 /114 7\°-74 • 

PjVf'K = P E V E ^ or Vj= (&) V E = [^Q 0.0146 D = 0.0318 2). 

/ p \ 0.74 /I 14 7\ °."4 

P/Fi i.35 = p E F ^i.35 or F/ = / Le \ v E =[ i^j ) 0.0146 2) = 0.0324 2). 



COMPRESSOR DESIGN 117 

The piston displacement is V z — V E — 0.3494 D; the volumetric efficiency is the quo- 
tient of (V K - V ) = 0.3256 D by this displacement, or 0.982. For a stroke of 4 ft., 
the cylinder diameter is V[(0.3494 D = 1547.84) - 640] x 144 -=- 0.7854 = 21.06 in. 
The power consumption for this stage is 

W - 144 2)r ni7 d . (114.7 x 0.1642) -(39.14 x 0.364) 

33000 L v ' ' J 0.35 

- (39.14 x 0.3316) - (114-7 x 0.0146)^(39.14 x 0.0324)j 
= 315.08 horse power. 
The total horse power for the two-stage compressor is then 630.33, and (within 
the limit of the error of computation) the work is equally divided between the 
stages. 

235. Comparisons. We note then, that in two-stage compression, the saving 

in power is ^ — = 0.118 of the power expended in single-stage compres- 

715 

sion; that the low-pressure cylinder of the two-stage machine is somewhat smaller 

than the cylinder of the single-stage compressor; and that, in the two-stage 

machine, the cylinder areas are (approximately) inversely proportional to tlie suction 

pressures. The amount of cooling water required will be found to be several times 

that necessary in the single-stage compressor. 

236. Power Plant Applications. On account of the ease of solution of air in 
water, the boiler feed and injection waters in a power plant always carry a con- 
siderable quantity of air with them. The vacuum pump employed in connection 
with a condenser is intended to remove this air, as well as the water. It is esti- 
mated that the waters ordinarily contain about Jq of their volume of air at atmos- 
pheric pressure. The pump must be of size sufficient to handle this air when 
expanded to the pressure in the condenser. Its cycle is precisely that of any air 
compressor, the suction stroke being at condenser pressure and the discharge 
stroke at atmospheric pressure. The water present acts to reduce the value of the 
exponent n, thus permitting of fair economy. 

237. Dry Vacuum Pumps. In some modern forms of high vacuum apparatus, 
the air and water are removed from the condenser by separate pumps. The 
amount of air to be handled cannot be computed from the pressure and tempera- 
ture directly, because of the water vapor with which it is saturated. From Dal- 
ton's law, and by noting the temperature and pressure in the condenser, the pressure 
of the air, separately considered, may be computed. Then the volume of air, cal- 
culated as in Art. 236, must be reduced to the condenser temperature and pressure, 
and the pump made suitable for handling this volume (14). 

Commercial Types of Compressing Machinery 

238. Classification of Compressors. Air compressors are classified according 
to the number of stages, the type of frame, the kind of valves, the method of 
driving, etc. Steam-driven compressors are usually mounted as one unit with the 
steam cylinders and a single common fly wheel. Regulation is usually effected by 



118 



APPLIED THERMODYNAMICS 




TYPES OF COMPRESSOR 



119 



varying the speed. The ordinary centrifugal governor on the steam cylinder im- 
poses a maximum speed limit; the shaft governor is controlled by the air pressure, 
which automatically changes the point of cut-off on the steam cylinder. Power- 
driven compressors may be operated by electric motor, belt, water wheel, or in 
other ways. They are usually regulated by means of an " unloading valve," which 
either keeps the suction valve closed during one or more strokes or allows the air 
to discharge into the atmosphere whenever the pipe lines are fully supplied. In 
air lift practice, a constant speed is sometimes desired, irrespective of the load. 
In the "variable volume" type of machine, the delivery of the compressor is 
varied by closing the suction valve before the completion of the suction stroke. 
The air in the cylinder then expands below atmospheric pressure. 



239. Standard Forms. The ordinary small compressor is a single-stage 
machine, with poppet air valves on the sides of the cylinder. The frame is of the 
"fork" pattern, with bored guides, or of the " duplex " type, with two single-stage 
cylinders. These machines maybe either steam or belt driven. The "straight 
line " compressors differ from the duplex in having all of the cylinders in one 
straight line, regardless of their number. 

For high-grade service, in large units, the standard form is the cross-compound 
two-stage machine, the low-pressure steam and air cylinders being located tandem 
beside the high -pressure cylinders, and the air cylinders being outboard, as in 
Fig. 86. Ordinary standard machines of this class are built in capacities ranging 
up to 6000 cu. ft. of free air per minute. The other machines are usually con- 
structed only in smaller sizes, ranging down to as small as 100 cu. ft. per minute. 

Some progress has been made in the development of rotary compressors for 
direct driving by 

steam turbines. The I ^ If/ ^B^ ; Y/ „ ' ' ' ^_ ! 

efficiency is fully as 
high as that of an 
ordinary reciprocat- 
ing compressor, and 
the mechanical losses 
are much less. 

240. Hydraulic 
Piston Compressors; 
Sommeiller's. In Fig. I 
87, as the piston F 
moves to the right, 
air is drawn through 
C to G, to- 
gether with 
cooling water 
from B. On 
the return 



n 




Fig. 87. Art. 240. — Sommeiller Hydraulic Piston Compressor. 

Indicator dia- 



stroke, the air is compressed and discharged through D and A. 
grams are given in Fig. 88. 



120 



APPLIED THERMODYNAMICS 



The value of n is exceptionally low, and clearance expansion almost elimi- 
nated. This was the first commercial piston compressor, and it is still used to a 




Fig. 88. Art. 240. — Variable Discharge Pressure Indicator Diagrams, Sommeiller 

Compressor. 

limited extent in Europe, the large volume of water present giving effective cool- 
ing. It cannot be operated at high speeds, on account of the inertia of the 
water. 

The Leavitt hydraulic piston compressor at the Calumet and Hecla copper 
mines, Michigan, has double-acting cylinders 60 by 42 in., and runs at 25 revolu- 
tions per minute, a comparatively 
high speed. The value of n from the 
card shown in Fig. 89 is 1.23. 

241. Taylor Hydraulic Compressor. 

Water is conducted through a vertical 
shaft at the necessary head (2.3 ft. per 
pound pressure) to a separating cham- 




Fig. 89. 



Art. 248. — Cards from Leavitt 
Compressor. 



Fig. 90. Art. 241. — Taylor Hydraulic 
Compresssor. 



TYPES OF COMPRESSOR 



121 



ber. The shaft is lined with a riveted or cast-iron cylinder, and at its top is a 
dome, located so that the water flows downward around the inner circumference 
of the cylinder. The dome is so made that the water alternately contracts and 
expands during its passage, producing a partial vacuum, by means of which air is 
drawn in through numerous small pipes. The air is compressed at the tempera- 
ture of the water while descending the shaft. The separating chamber is so 
large as to permit of separation of the air under an inverted bell, from which it is 
led by a pipe. The efficiency is 0.60 to 0.70, some air being always carried away 
in solution. The initial cost is high, and the system can be installed only where 
a head of water is available. Figure 90 illustrates the device (15). 

242. Details of Construction. The standard form of cylinder for large machines 
is a two-piece casting, the working barrel being separate from the jacket, so that 
the former may be a good wearing metal and may be quite readily removable. 
Access to the jacket space is provided through bolt holes. 

On the smaller compressors, the poppet type of valve is frequently used for both 
inlet and discharge (Fig. 91). It is usually considered best to place these valves 




Fig. 91. 



Art. 242. — Compressor Cylinder with Poppet Valves. 

(Clayton Air Compressor Works.) 



in the head, thus decreasing the clearance. They are satisfactory valves for auto- 
matically controlling the point of discharge, excepting that they are occasionally 
noisy and uncertain in closing. Poppet valves work poorly at very low pressures, 
and are not generally used for controlling the intake of air. Some form of 
mechanically operated valve is preferably employed, such as the semi-rocking type 



122 



APPLIED THERMODYNAMICS 



DISCHARGE. 



of Fig. 92, located at the bottom of the cylinder, which has poppet valves for the 
discharge at the top. For large units, Corliss inlet valves are usually employed, 

these being rocking cyl- 
indrical valves running 
crosswise. As in steam 
engines, they are so 
driven from an eccentric 
and wrist plate as to give 
rapid opening and closing 
of the port, with a com- 
paratively slow interven- 
ing movement. They are 
not suitable for use as 
discharge valves in single- 
stage compressors, or in 
the high-pressure cylin- 
ders of multi-stage com- 
pressors, as they become 
fully open too late in the 
stroke to give a suffi- 
ciently free discharge. 
In Fig. 93 Corliss valves 
are used for both inlet 
and discharge. The 
auxiliary poppet shown 
is used as a safety valve. 
A gear sometimes used consists of Corliss inlet valves and mechanically operated 
discharge valves, which, latter, though expensive, are free from the disadvantages 
sometimes experienced with poppet valves. The closing only of these valves is 
mechanically controlled. Their opening is automatic. 




Fig. 



SUCTION 

Art. 242. — Compressor Cylinder with Rocking Inlet 
Valves. (Clayton Air Compressor Works.) 




Fig. 93. Art. 242. — Compressor Cylinder with Corliss Valves. (Allis-Chalmers Co.) 



COMPRESSED AIR TRANSMISSION 123 



Compressed Air Transmission 

243. Transmissive Losses. The air falls in temperature and pressure in the 
pipe line. The fall in temperature leads to a decrease in volume, which is further 
reduced by the condensation of water vapor ; the fall in pressure tends to increase 
the volume. Early experiments at Mont Cenis led to the empirical formula 
F = 0.00000936 (?i 2 l + d), for a loss of pressure F in a pipe d inches in diameter, 
I ft. long, in which the velocity is n feet per second (16). 

In the Paris distributing system, the main pipe was 300 mm. in diameter, and 
about f in. thick, of plain end cast iron lengths connected with rubber gaskets. 
It was laid partly under streets and sidewalks, and partly in sewers, involving the 
use of many bends. There were numerous drainage boxes, valves, etc., causing 
resistance to the flow ; yet the loss of pressure ranged only from 3.7 to 5.1 lb., the 
average loss at 3 miles distance being about 4.4 lb., these figures of course including 
leakage. The percentage of air lost by leakage was ascertained to vary from 0.38 
to 1.05, including air consumed by some small motors which were unintentionally 
kept running while the measurements were made. This loss would of course be 
proportionately much greater when the load was light. 

244. Unwin's Formula. Unwin's formula for terminal pressure after long 
transmission is generally employed in calculations for pipe lines (17). It is, 



P L 430 T d\ 



in which p = terminal pressure in pounds per square inch, 
P = initial pressure in pounds per square inch, 
/= an experimental coefficient, 
u — velocity of air in feet per second, 
L = length of pipe in feet, 
d = diameter of (circular) pipe in feet, 
T = absolute temperature of the air, F°. 

A simple method of determining/is to measure the fall of pressure under known 
conditions of P, u, T, L, and d, and apply the above formula. Unwin has in this 
way rationalized the results of Riedler's experiments on the Paris distributing 
system, obtaining values ranging from 0.00181 to 0.00449, with a mean value 
/= 0.00290. For pipes over one foot in diameter, he recommends the value 0.003 ; 
for 6-inch pipe,/= 0.00435; for 8-inch pipe,/= 0.004. 

Riedler and Gutermuth found it possible to obtain pipe lengths as great as 
10 miles in their experiments at Paris. Previous experiments had been made, on 
a smaller scale, by Stockalper. For cast-iron pipe, a harmonization of these 
experiments gives /= 0.0027(1 -f 0.3 c?), d being the diameter of the pipe in feet. 
The values of f for ordinary wrought pipe are probably not widely different. In 
any well-designed plant, the pressure loss may be kept very low. 

245. Storage of Compressed Air. Air is sometimes stored at very high pres- 
sures for the operation of locomotives, street cars, buoys, etc. An important con- 



124 APPLIED THERMODYNAMICS 

sequence of the principle illustrated in Joule's porous plug experiment (Art. 74) 
here comes into play. It was remarked in Art. 74 that a slight fall of temperature 
occurred during the reduction of pressure. This was expressed by Joule by the 
formula 



0.92 



in which F was the fall of temperature in degrees Centigrade for a pressure 
drop of 100 inches of mercury when T was the initial absolute temperature 
(Centigrade) of the air. For air at 70° F., this fall is only 1|° F., but when stored 
air at high pressure is expanded through a reducing valve for use in a motor, the 
pressure change is frequently so great that a considerable reduction of tempera- 
ture occurs. The efficiency of the process is very low ; Peabody cites an instance 
(18) in which with a reservoir of 75 cu. ft. capacity, carrying 450 lb. pressure, 
with motors operating at 50 lb. pressure and compression in three stages, the 
maximum computed plant efficiency is only 0.29. An element of danger arises in 
compressed air storage plants from the possibility of explosion of minute traces 
of oil at the high temperatures produced by compression. 

246. Liquefaction of Air ; Linde Process (19). The fall of temperature accom- 
panying a reduction of pressure has been utilized by Linde and others in the 
manufacture of liquid air. Air is compressed to about 2000 lb. pressure in a 
three-stage machine, and then delivered to a cooler. This consists of a double 
tube about 400 ft. long, arranged in a coil. The air from the compressor passes 
through the inner tube to a small orifice at its farther end, where it expands into 
a reservoir, the temperature falling, and returns through the outer tube of the 
cooler back to the compressor. At each passage, a fall of temperature of about 
37|° C. occurs. The effect is cumulative, and the air soon reaches a temperature 
at which the pressure will cause it to liquefy (Art. 610). 

247. Refrigeration by Compressed Air. This subject will be more particularly 
considered in a later chapter. The fall of temperature accompanying expansion 
in the motor cylinder, with the difficulties which it occasions, have been men- 
tioned in Art. 185. Early in the Paris development, this drop of temperature was 
utilized for refrigeration. The exhaust air was carried through flues to wine 
cellars, where it served for the cooling of their contents, the production of ice, etc. 
In some cases, the refrigerative effect alone is sought, the performance of work 
during the expansion being incidental. 

(1) Riedler, Neue Erfahrungen ilber die Kraftversorgung von Paris durch Druck- 
luft: Berlin, 1891. (2) Pernolet (V Air Comprime) is the standard reference on this 
work. (3) Experiments upon Transmission, etc. (Idelled.), 1903, 98. (4) Unwin, op. 
cit., 18 et seq. (5) Unwin, op. cit., 32. (6) Graduating Thesis, Stevens Institute of 
Technology, 1891. (7) Unwin, op. cit., 48. (8) Op. cit, 109. (9) Unwin, op. cit., 48, 49 ; 
some of the final figures are deduced from Kennedy's data. (10) Power, February 23, 
1909, p. 382. (11) Development and Transmission of Power, 182. (12) Engineering 
News, March 19, 1908, 325. (13) Peabody, Thermodynamics, 1907, 378. (14) Ibid., 
375. (15) Hiscox, Compressed Air, 1903, 273. (16) Wood, Thermodynamics, 1905, 
306. (17) Transmission by Compressed Air, etc., 68 ; modified as by Peabody. 



COMPRESSED AIR 125 

(18) Thermodynamics, 1907, 393, 394. (19) Zeuner, Technical Thermodynamics 
(Klein) ; II, 303-313 : Trans. A. S. M. E., XXI, 156. 



SYNOPSIS OF CHAPTER IX 

The use of compressed cold air for power engines and pneumatic tools dates from 1860. 

The Air Engine 

The ideal air engine cycle is bounded by two constant pressure lines, one constant 
volume line, and a polytropic. In practice, a constant volume drop also occurs 
after expansion. 

Work formulas : 

pv+pvloge^-qV; pv + pv ~ ? V - q V; pvlog e -; (pv-PV)f^-)- 
v n — 1 * ° v \y — 1) 

Preheaters prevent excessive drop of temperature during expansion ; the heat em- 
ployed is not wasted. 

Cylinder volume = 33,000 NBt -f 2w Up, ignoring clearance. 

To ensure quiet running, the exhaust valve is closed early, the clearance air acting as a 
cushion. This modifies the cycle. 

Early closing of the exhaust valve also reduces the air consumption. 

Actual figures for free air consumption range from 400 to 2400 cu. ft. per Ihp-hr. 

The Compressor 

The cycle differs from that of the engine in having a sharp "toe" and a complete clear- 
ance expansion curve. 

Economy depends largely on the shape of the compression curve. Close approximation 
to the isothermal, rather than the adiabatic, should be attained, as during expan- 
sion in the engine. This is attempted by air cooling, jet and spray injection of 
water, and jacketing. Water required^ C—H-^ (S—s) . 

HCheat to be abstracted) = -^— X l^-Y^ 1 - 1~| + -^- —2Z-(*)i 
n-lL\PI J y — 1 y—\\p) 

Multi-stage operation improves the compression curve most notably and is in other 

respects beneficial. 
Intercooling leads to friction losses but is essential to economy ; must be thorough. 

Work, neglecting clearance (single cylinder) ,= W= — ^— ( 2- 

The area under the compression curve is called the work of compression. 
Minimum work, in two-stage compression, is obtained when P 2 = qp. 



Engine and Compressor Belations 

Compressive efficiency : ratio of engine work to compressor work ; 0.5 to 0.9. 
Mechanical efficiency : ratio of work in cylinder and work at shaft ; 0.80 to 0.90. 
Cylinder efficiency : ratio of ideal diagram area and actual diagram area ; 0.70 to 0.90. 
Plant efficiency : ratio of work delivered by air engine to work expended at compressor 
shaft ; 0.25 to 0.45 ; theoretical maximum, 1.00. 



L m\P) mj 



126 APPLIED THERMODYNAMICS 

The combined ideal entropy diagram is bounded by two constant pressure curves and 
two polytropics. The economy of thorough intercooling with multi-stage operation 
is shown ; as is the importance of a low exponent for the polytropics. With very 
cold water, the net power consumption might be negative. 

Compressor Capacity 

Volumetric efficiency • = ratio of free air drawn in to piston displacement ; it is decreased 
by excessive clearance, suction friction, heating during suction, and installation at 
high altitudes. Long stroke compressors have^ proportionately less clearance. 
Water may be used to fill the clearance space : multi-stage operation makes 
clearance less detrimental ; refrigeration of the entering air increases the volumet- 
ric efficiency. Its value ranges ordinarily from 0.70 to 0.92. Suction friction 
and clearance also decrease the cylinder efficiency, as does discharge friction. 

Compressor Design 

Theoretical piston, displacement per stroke = -^ — , or including clearance, 

pvT 
2nPt 

to be increased 5 to 10 per cent in practice. 
In a multi-stage compressor with perfect intercooling, the cylinder volumes are inversely 

as the suction pressures. 
The power consumed in compression may be calculated for any assumed compressive 

path. 
A typical problem shows a saving of 12 per cent by two-stage compression. 
The " vacuum pump" used with a condenser is an air compressor. 

Commercial Types of Compressing Machinery 

Classification is by number of stages, type of frame or valves, or method of driving. 
Governing is accomplished by changing the speed, the suction, or the discharge pressure. 
Commercial types include the single, duplex, straight line and cross-compound two-stage 

forms, the last having capacities up to 6000 cu. ft. per minute. Some progress has 

been made with turbo-compressors. 
Hydraulic piston compressors give high efficiency at low speeds. 
The Taylor hydraulic compressor gives efficiencies up to 0.60 or 0.70. 
Cylinder barrels and jackets are separate castings. Access to water space must be 

provided. 
Poppet, mechanical inlet, Corliss, and mechanical discharge valves are used. 

Compressed Air Transmission 

Loss in pressure = 0.00000936 nH -=- d. 

In Paris, the total loss in 3 miles, including leakage, was 4-4 lb. ; the percentage of leak- 
age was 0.38 to 1.05, including air unintentionally supplied to consumers. 

Unwin's formula; p = p\l - -^-^ \l Mean value of /= 0.0029. /= 0.0027(1 +0.3 d). 

Fall of temperature for 49 lb. fall of pressure by throttling = 0.92 I zl2iL J . 



COMPRESSED AIR 127 

Stored high pressure air may be used for driving motors, but the efficiency is low. 
The fall of temperature induced by throttling may be used cumulatively to liquefy air. 
The fall of temperature accompanying expansion in the engine may be employed for 
refrigeration. 

, PROBLEMS 

1. An air engine works between pressures of 180 lb. and 15 lb. per square inch, 
absolute. Find the work done per cycle with adiabatic expansion from v = 1 to V = 4, 
ignoring clearance. By what percentage would the work be increased if the expansion 
curve were PV 1 - 3 = c? 

2. The expansion curve is PV h3 = c, the pressure ratio during expansion 7 : 1, the 
initial temperature 100° F. Find the temperature after expansion. To what tempera- 
ture must the entering air be heated if the final temperature is to be kept above 32° F ? 

3. Find the cylinder dimensions for a double-acting 100 hp. air engine with clear- 
ance 4 per cent, the exhaust pressure being 15 lb. absolute, the engine making 200 
r. p. m., the expansion and compression curves being PF U5 = c, and the air being 
received at 160 lb. absolute pressure. Compression is carried to the maximum pres- 
sure, and the piston speed is 400 ft. per minute. A 10-lb. drop of pressure occurs at 
the end of expansion. (Allow a lOper cent margin over the theoretical piston dis- 
placement.) 

4. Estimate the free air consumption per Ihp.-hr. in the engine of Problem 3. 

5. A hydrogen compressor receives its supply at 70° F. and atmospheric pressure, 
and discharges it at 100 lb. guage pressure. Find the temperature of discharge, if the 
compression curve is PV 1 - 32 = c. 

6. In Problem 5, what is the percentage of power wasted as compared with iso- 
thermal compression, the cycles being like CBAD, Fig. 57 ? Consider only the power 
necessary to compress isothermally to the maximum pressure, not the whole power 
expended in the cycle. 

7. In Problem 3, find what quantity of heat must have been added during expan- 
sion to make the path PV 1 - 35 — c rather than an adiabatic. Assuming this to be added 
by a water jacket, the water cooling through a range of 70°, find the weight of water 
circulated per minute. 

8. Find the receiver pressures for minimum work in two and four-stage compres- 
sion of atmospheric air to guage pressures of 100, 125, 150, and 200 lb. 

9. What is the minimum, work expenditure in the cycle compressing free air at 
70° F. to 100 lb. guage pressure, per pound of air, along a path PV 1 - 35 = c, clearance 
being ignored ? 

10. Find the cylinder efficiency in Problem 3, the pressure in the pipe line being 
165 lb. absolute. 

11. Sketch the entropy diagram for a four-stage compressor and two-stage air 
engine, in which n is 1.3 for the compressor and 1.4 for the engine, the air is inade- 
quately intercooled, perfectly aftercooled, and inadequately preheated between the 
engine cylinders. Compare with the entropy diagram for adiabatic paths and perfect 
intercooling and such preheating as to keep the temperature of the exhaust above 32° F. 



128 APPLIED THERMODYNAMICS 

12. Find the cylinder dimensions and power consumption of a single-acting single- 
stage air compressor to deliver 8000 cu. ft. of free air per minute at 180 lb. absolute 
pressure at 60 r. p. m., the intake air being at 13.0 lb. absolute pressure, the piston speed 
640 ft. per minute, clearance 3 per cent, and the expansion and compression curves fol- 
lowing the law PV IS1 = c. 

13. With conditions as in Problem 12, find the cylinder dimensions and power 
consumption if compression is in two stages, intercooling is perfect,and 2 lb. of friction 
loss occur between the stages. 

14. The cooling water rising from 68° F. to 89° F. in temperature, in Art. 233, 
find the water consumption in gallons per minute. 

15. Find the water consumption for jackets and intercooling in Art. 234, the range 
of temperature of the water being from 47° to 68° F. 

16. Find the cylinder volume of a pump to maintain 26" vacuum when .pumping 
100 lb. of air per minute, the initial temperature of the air being 110° F., compression 
and expansion curves PF 1 - 28 = c, clearance 6 per cent., and the pump having two 
double-acting cylinders. The speed is 60 r. p. m. Pipe friction may be ignored. 

17. Compare the Riedler and Gutermuth formula for / (Art. 244) with Unwin's 
values. 

18. In a compressed air locomotive, the air is stored at 2000 lb. pressure and de- 
livered to the motor at 100 lb. Find the temperature of the air delivered to the motor 
if that of the air in the reservoir is 80° F., assuming that the value of F (Art. 245) is 
directly proportional to the pressure drop. 

19. With isothermal curves and no friction, transmission loss, or clearance, what 
would he the combined efficiency from compressor to motor of an air storage system in 
which the storage pressure was 450 lb. and the motor pressure 50 lb.? The tempera- 
ture of the air is 80° F. at the motor reducing valve. 

20. Find, by the Mont Cenis formula, the loss of pressure in a 12-in. pipe 2 miles 
long in which the air velocity is 32 ft. per second. Compare with Un win's formula, 
using the Riedler and Gutermuth value for/, assuming P = 80, T = 70° F. 



CHAPTER X 

HOT-AIR ENGINES 

248. General Considerations. From a technical standpoint, the class of 
air engines includes all heat motors using any permanent gas as a working 
substance. For convenience, those engines in which the fuel is ignited 
inside the cylinder are separately discussed, as internal combustion or gas 
engines (Chapter XI). The air engine proper is an external combustion 
engine, although in some types the products of combustion do actually 
enter the cylinder ; a point of disadvantage, since the corrosive and gritty 
gases produce rapid wear and leakage. The air engine employs, usually, 
a constant mass of working substance, i.e. the same body of air is alter- 
nately heated and cooled, none being discharged from the cylinder and no 
fresh supply being brought in ; though this is not always the case. Such 
an engine is called a " closed " engine. Any fuel may be employed ; the 
engines require little attention ; there is no danger of explosion. 

Modern improvements on the original Stirling and Ericsson forms of 
air engine, while reducing the objections to those types, and giving excel- 
lent results in fuel economy, are, nevertheless, limited in their application 
to small capacities, as for domestic pumping service. 

In air, or any perfect gas, the temperature may be varied independ- 
ently of the pressure ; consequently, the limitation referred to in Art. 143 
as applicable to steam engines does not necessarily apply to air engines, 
which may work at much higher initial temperatures than any steam en- 
gine, their potential efficiency being consequently much greater. When 
a specific cycle is prescribed, however, as we shall immediately find, pres- 
sure limits may become of importance. 

249. Capacity. One objection to the air engine arises from the ex- 
tremely slow transmission of heat through metal surfaces to dry gases. 
This is partially overcome in various ways, but the still serious objection 
is the small capacity for a given size. If the Carnot cycle be plotted for 
one pound of air, as in Fig. 94, the enclosed work area is seen to be very 
small, even for a considerable range of pressures. The isothermals and 
adiabatics very nearly coincide. For a given output, therefore, the air en- 
gine must be excessively large at anything like reasonable maximum pres- 
sures. In the Ericsson engine (Art. 269), for example, although the cycle 

129 



130 



APPLIED THERMODYNAMICS 



was one giving a larger work area than that of Carnot, four cylinders 
were required, each having a diameter of 14 ft. and a stroke of 6 ft. ; it 
was estimated that a steam engine of equal power would have required 



150 



ll 


























1 






















































































































* 












4- 




























\ 


























\\ 


























\ 


























\ 




T = l 


D59.6 a 


bs. 
















959.6 


abs. 


— t^. 






































I? 




























= w 










— — 3 



" 12 3 4 5 6 7 8 9 10 11 12 13 

Fig. 94. Arts. 249, 250. — Carnot Cycle for Air. 

only a single cylinder, 4 ft. in diameter and of 10-ft. stroke, running at 17 
revolutions per minute and using 4 lb. of coal per horse power per hour. 
The air engine ran at 9 r. p. m., and its great bulk and cost, noisiness and 
rapid deterioration, overbore the advantage of a much lower fuel consump- 
tion, 1.87 lb. of coal per hp.-hr. At the present time, with increased 
steam pressures and piston speeds, the equivalent steam engine would be 
still smaller. 



250. Carnot Cycle Air Engine. The efficiency of the cycle shown in 
Fig. 94 has already been computed as (T—t)+ T (Art. 135). The work 
done per cycle is, from Art. 135, 

TT=j?^riog.^-no g .^ = iJ(r-oiog.^ = 22(r-oiog.-S- 



POLYTROPIC CYCLE 



131 



Another expression for the work, since 



^= -\ is W=R(T-t) log e 5. 



But from Art. 104, £*'=[T)v- : 



Fifty- 1 



P* = ^(jT 1 and TF= JR(!T- Q log. ^ 

This can have a positive value only when — I ( — V -1 exceeds unity ; which 

Pz\T) 



p 

is possible only when — - exceeds 



Now since P l and P 3 are the 



limiting pressures in the cycle, and since for air y -i- (y — 1) = 3.486, the 
minimum necessary ratio of pressures increases as the 3.486 power of the ratio 
of temperatures.* This alone makes the cycle impracticable. In Pig. 94, 
the pressure range is from 14.7 to 349.7 lb. per square inch, although the 
temperature range is only 100°. 



251. Polytropic Cycle. In Fig. 95, let T, t be two isothermals, eb and df two 
like polytropic curves, following the law pv n = c, and ed and bf two other like 
polytropic curves, following the law pv m = c. 
Then ebfd is a polytropic cycle. Let T, t, P b , P e 



be given. Then T e = T 



T 

6 1 




\y x| ;n 




Fig. 96. Arts. 251, 256. 
tropic Cycle. 



Poly- 



. In the en- 
tropy diagram, 
Fig. 96, locate the 
isothermals T, t, 
T e . Choose the 
point e at random. 
From Art. Ill, the 
specific heat along 
a path pv n = c is 

s=; fe) ;and 

from Art. 163, the increase of entropy when the 
specific heat is s, in passing from e to b, is 

T 

N — s log e — . This permits of plotting the curve 

J- e 



Fig. 95. Arts. 251, 256, Prob. 4a. 
— Polytropic Cycle. 



* It has been shown that — 
P 4 






But P 3 < P 4 , if a finite work area is to 



be obtained ; hence — 



Pi 

Ps 






132 



APPLIED THERMODYNAMICS 



eb in successive short steps, in Fig. 96. Along ed, similarly, s, = l(™ U. ) and 

T \m - 1/ 

N x = s 1 log e — between d and e. We complete the diagram by drawing bf and 

df, establishing the point of intersection which determines the temperature at /. 



We find T f :T b ::T d : T e . The efficiency is equal to 
[nebx + xbfN - ydfN - nedy\ V [nebx + xbfN~\ 



ebfd 
nebfN 



, or to 



= 1 



(T - T e ) + Sl (T b - T f ) - s(T f - T d ) - s x (T e 
s(T b - T e ) + Sl (n- Tf) 

s(T f - T^ + s^T,- T (l ) 
s(T b -T)+ Sl {T b -T f y 



Ta) 



the negative sign of the specific heat s x being disregarded. 

252. Lorenz Cycle. In Fig. 97 let dg and bh be adiabatics, and let the curves 
gb and dh be poly tropics, but unlike, the former having the exponent n, and the 
latter the exponent q. This constitutes the cycle of Lorenz. We find the tempera- 




Fig. 97. Arts. 252, 256, Prob. 5.— 
Lorenz Cycle. 




Fig. 98. Arts. 252, 256.— Lorenz Cycle, 
Entropy Diagram. 



ture at g as in Art. 251, and in the manner just described plot the curves gb and 
dh on the entropy diagram, Fig. 98, P g , P b , T b , T d , n and q being given, dg and 
bh of course appear as vertical straight lines. The efficiency is 

s n (T b - T g ) -s g (T h - T d ) 
s n (T b -T g ) 



253. Reitlinger Cycle. This appears as aicj, Figs. 99 and 100. It is bounded 
by two isothermals and two like polytropics (isodiabatics). To plot the entropy 
diagram, Fig- 100, we assume the ratio of pressures or of volumes along ai or cj. 

Let V a and V* ^ e given. Then the gain of entropy from a to i is ( P a V a log e — ?' ) -r- T. 



JOULE AIR ENGINE 



133 



The curves ic and aj are plotted for the given value of the exponent n. This is 
sometimes called the isodiabatic cycle. Its efficiency is 

(Hai + H ic — H jc — H aj ) -f- (H (li + H ic ), 

which may be expanded as in Arts. 251, 252. 




Fig. 99. Arts. 253, 256. 
linger Cycle. 



Reit- 




F.g. 100. Arts. 253, 256, 257, 258 
259. — Reitlinger Cycle, Entropy 
Diagram. 



254. Joule Engine. An air engine proposed by Ericsson as early as 
1833, and revived by Joule and Kelvin in 1851, is shown in Fig. 101. A 
chamber C contains air kept at a low temperature t by means of circulating 
water. Another chamber A contains hot air in a state of compression, 
the heat being supplied at a constant temperature T by means of an ex- 
ternal furnace (not shown). Mis a pump cylinder by means of which, air 





,1 


i 


ENGINE 

N 






llli 




u 










\ 1 


,ra_ 




111 














WZ 






HOT CHAMBER <-"-. 

V 


111 


M 






pi 


MP 





COLO 
CHAMBER 



Fig. 101. Arts. 254, 255, 275.— Joule Air Engine. 



may be delivered from C to A, and N is an engine cylinder in which air 
from A may be expanded so as to perform work. The chambers A and C 
are so large in proportion to M and N that the pressure of the air in these 
chambers remains practically constant. 



134 



APPLIED THERMODYNAMICS 



The pump M takes air from C, compresses it adiabatically, until its 
pressure equals that in A, then, the valve v being opened, delivers it to A 

at constant pressure. The cycle 
is fdoe, Fig. 102. In this special 
modification of the polytropic 
cycle of Art. 251, fd represents 
the drawing in of the air at con- 
stant pressure, do its adiabatic 
compression, and oe its discharge 
to A. Negative work is done, 
equal to the area fdoe. Concur- 
rently with this operation, hot 
air has been flowing from A to N 
through the valve u, then expand- 
ing adiabatically while u is closed ; finally, when the pressure has fallen 
to that in C, being discharged to the latter chamber, the cycle being ebqf, 
Fig. 102. Positive work has been done, and the net positive work per- 
formed by the whole apparatus is ebqf — fdoe — obqd. 




Fig. 102. Arts. 254, 255, 256.— Joule Cycle. 



255. Efficiency of Joule Engine. We will limit our attention to the net 
cycle obqd. The heat absorbed along the constant pressure line ob is 
H* = k(T - T ). The heat rejected along qd is H qd = k(T q - t). But 

^ , and the efficiency is 



T T T 

from Art. 251, -f = — , whence, ^_ 



H 



H qd 



H n 



= 1 



Hqd 

H oh 



! T *-t 



T a ~t 




This is obviously less than the 
efficiency of the Carnot cycle 
between T and t. The entropy 
diagram may be readily drawn 
as in Fig. 103. The atmos- 
phere may of course take the 
place of the cold chamber C, 
a fresh supply being drawn in 
by the pump at each stroke, and 
the engine cylinder likewise 
discharging its contents to the 
atmosphere. The ratio fd -i- fq, 
in Fig. 102, shows the necessary ratio of volumes of pump cylinder and 
engine cylinder. The need of a large pump cylinder would be a serious 
drawback in practice ; it would make the engine bulky and expensive, and 



Fig. 103. 



Arts. 255, 25(5.— Joule Cycle, Entropy 
Diagram. 



REGENERATOR 



135 



would lead to an excessive amount of mechanical friction, 
ensrine has never been constructed. 



The Joule 



p 

a 




~ T 


e °\\ 


JL/t= C \ -~""~ 




\ <J \\ 

w 


\ ^°^N 


LA \ c 


d 


j 


L_££=__c_^c 






\ 



256. Comparisons. The cycles just described have been grouped 
in a single illustration in Fig. 104. Here we have, between the 
temperature limits T and £, the Carnot cycle, abed ; the polytropic 
cycle, debf; the Lorenz 
cycle, dgbh ; that of Reit- 
linger, aiej ; and that of 
Joule, obqd. These illus- 
trations are lettered to 
correspond with Figs. 
95-100, 102, 103. A 
graphical demonstration 
that the Carnot cycle is 
the one of maximum 
efficiency suggests itself. 
We now consider the 
most successful attempt 
yet made to evolve a cycle 
having a potential effi- 
ciency equal to that of 
Carnot. 

257. Regenerators. 

By reference to Fig. 100, 

it may be noted that the 

heat areas under aj and 

ic are equal. The heat 

absorbed along the one 

path is precisely equal to 

that rejected along the 

other. This fact does 

not prevent the efficiency 

from being less than that 

of the Carnot cycle, for 

efficiency is the quotient 

of work done by the gross 

heat absorption. If, however, the heat under ic were absorbed 

not from the working substance, and that under ja were rejected 




Fig. 104. Arts. 256, 266. — Hot-air Cycles. 



136 APPLIED THERMODYNAMICS 

not to the condenser ; but if some intermediate body existed having a 
storage capacity for heat, such that the heat rejected to it along ja 
could be afterward taken up from it along ic, then we might ignore 
this quantity of heat as affecting the expression for efficiency, and the 
cycle would be as efficient as that of Carnot. The intermediate body 
suggested is called a regenerator. 



258. Action of Regenerators. Invented by Robert Stirling about 1816, and 
improved by James Stirling, Ericsson, and Siemens, the present form of regener- 
ator may be regarded as a long pipe, the walls of which have so large a capacity 
for heat that the temperature at any point remains practically constant. Through 
this pipe the air flows in one direction when working along ic, Fig. 100, and 
in the other direction while working along ja. The air encounters a gradually 
changing temperature as it traverses the pipe. 

Let hot exhaust air, at i, Fig. 100, be delivered at one end of the regenerator. 
Its temperature begins to fall, and continues falling, so that when it leaves the 
regenerator its temperature is that at c, usually the temperature of the atmosphere. 
The temperature at the inlet end of the regenerator is then T, that at its outlet t. 
During the admission of fresh air, along ja, it passes through the regenerator in 
the opposite direction, gradually increasing in temperature from t to T, without 
appreciably affecting the temperature of the re generator. Assuming the capacity of 
the regenerator to be unlimited, and that there are no losses by conduction of heat 
to the atmosphere or along the material of the regenerator itself, the process is 
strictly reversible. We may cause either the volume or the pressure to be either 
fixed or variable according to some definite law, during the regenerative move- 
ment. Usually, either the pressure or the volume is kept constant. 

As actually constructed, the regenerator consists of a mass of thin perforated 
metal sheets, so arranged as not to obstruct the flow of air. Some waste of heat 
always accompanies the regenerative process ; in the steamer Ericsson, it was 10 
per cent of the total heat passing through. Siemens appears to have reduced the 
loss to 5 per cent. 



259. Influence on Efficiency. Any cycle bounded by a pair of 
isothermals and a pair of like poly tropics, if worked with a regener- 
ator, has an efficiency ideally equal to that of the Carnot cycle. To 
be sure, the heated air is not all taken in at jP, nor all rejected at t; 
but the heat absorbed from the source is all at T, and that rejected 
to the condenser is all at t. The regenerative operations are mutu- 
ally compensating changes which do not affect the general principle 
of efficiency under such conditions. The heat paid for is only that 
under the line, at, Fig. 100. The regenerator thus makes the effi- 
ciency of the Carnot cycle obtainable by actual heat engines. 



THE STIRLING ENGINE 



137 



As will appear, the cycles in which a regenerator is commonly employed are 
not otherwise particularly efficient. Their chief advantage is in the large work 
area obtained, which means increased capacity of an engine of given dimensions. 
For highest efficiency, the regenerator must be added. 



260. The Stirling Engine. This important type of regenerative air engine 
was covered by patents dated 1827 and 1840, by Robert and James Stirling. Its 
action is illustrated in Fig. 105. G is the engine 
cylinder, containing the piston H, and receiving 
heated air through the pipe F from the vessel A A 
in which the air is alternately heated and cooled. 
The vessel AA is made with hollow walls, the inner 
lining being marked aa. The hemispherical lower 
portion of the lining is perforated ; while from A A 
up to CC the hollow space constitutes the regener- 
ator, being filled with strips of metal or glass. The 
plunger £ fits loosely in the machined inner shell 
aa. This plunger is hollow and filled with some 
non-conducting material. The spaces DD contain 
the condenser, consisting of a coil of small copper 
pipe, through which water is circulated by a sepa- 
rate pump. An air pump discharges into the pipe 
F the necessary quantity of fresh air to compensate 
for any leakage, and this is utilized in some cases 
to maintain a pressure which is at all stages con- 
siderably above that of the atmosphere. The furnace is built about the bottom 
ABA of the heating vessel. 




Fig. 105. 
263, 264. 



B 

Arts. 260, 261, 262, 

— Stirling Engine. 



261. Action of the Engine. Let the plunger E and the piston H be in their 
lowest positions, the air above E being cold. The plunger E is raised, causing 
air to flow from X downward through the regenerator to the space b, while H 
remains motionless. The air takes up heat from the regenerator, increasing its 
temperature, say to T, while the volume remains constant. After the plunger has come 
to rest, the piston H is caused to rise by the expansion produced by the absorption 
of heat from the furnace at constant temperature, the air reaching H by passing 
around the loose-fitting plunger E, which remains stationary. H now pauses in 
its "up" position, while E is lowered, forcing air through the regenerator from 
the lower space b to the upper space X, this air decreasing in temperature at con- 
stant volume. While E remains in its "down" position, H descends, forcing the 
air to the condenser D, the volume decreasing, but the temperature remaining con- 
stant at t. The cycle is thus completed. 

The working air has undergone four changes : (a) increase of pressure 
and temperature at constant volume, (6) expansion at constant tempera- 
ture, (c) a fall of pressure and temperature at constant volume, and (d) 
compression at constant temperature. 



138 



APPLIED THERMODYNAMICS 



262. Remarks. With action as described, the piston H and the plunger E 
(sometimes called the " displacer piston ") do not move at the same time ; one is 
always nearly stationary, at or near the end of its stroke, while the other moves. 
In practice, uniform rotative speed is secured by modifying these conditions, so 
that the actual cycle merely approximates that described. The vessel A A is 
sometimes referred to as the "receiver." It is obvious that a certain residual 
quantity of air is at all times contained in the spaces between the piston H and 
the plunger E. This does not pass through the regenerator, nor is it at any time 
subjected to the heat of the furnace. It serves merely as a medium for transmit- 
ting pressure from the "working air" to H\ and in contradistinction to that 
working substance, it is called " cushion air." Being at all times in communica- 
tion with the condenser, its temperature is constantly close to the minimum attained in 
the cycle. This is an important point in facilitating lubrication. 

263. Forms of the Stirling Engine. In some types, a separate pipe is carried 
from the lower part of the receiver to the working cylinder G, Fig. 105. This 
removes the necessity for a loose-fitting plunger ; in double-acting engines, each 
end of the cylinder is connected with the hot (lower) side of the one plunger and 
with the cold (upper) side of the other. In other forms, the regenerator has been 
a separate vessel ; in still others, the displacer plunger itself became the regen- 
erator, being perforated at the top and bottom and filled with wire gauze. The 
Laubereau-Schwartzkopff engine (1) is identical in principle with the Stirling, 
excepting that the regenerator is omitted. 

The maintenance of high minimum pressure, as described in Art. 260, while 
not necessarily affecting the efficiency, greatly increases the capacity, and (since 
friction losses are practically constant) the mechanical efficiency as well. 

P 




Fig. 100. Arts. 264, 265, 207. — Stirling Cycle. 



264. Pressure-Volume Diagram. The cycle of operations described in 
Art. 261 is that of Fig. 106, ABCD. Considering the cushion air, the 



THE STIRLING ENGINE 



139 



actual diagram which, would be obtained by measuring the pressures and 
volumes is quite different. Assume, for example, that the total volume 
of cushion air at maximum pressure (when E is at the top of its stroke 
and H is just beginning to move) is represented by the distance NE. 
Then if A I be laid off equal to NE, the total volume of air present is NI. 
Draw an isothermal EFHG, representing the path of the cushion air, sep- 
arately considered, while the temperature remains constant. Add its vol- 
umes, PF, ZH, QG, to those of working air, by laying off BK— PF, 
DM ' = ZH, CL=QG, at various points along the stroke. Then the 
cycle IKLM is that actually experienced by the total air, assuming the 
cushion air to remain at constant temperature throughout (Art. 262). 

The actual indicator diagrams obtained in tests are roughly similar to the 
cycle IKLM, Fig. 106; but the corners are rounded, and other distortions may 
appear on account of non-conformity with the ideal paths, sluggish valve action, 
errors of the indicating instrument, and various other causes. 



265. Efficiency. The heat absorbed from the source along AB, Fig. 
106, is P A V A \og e —^- That rejected to the condenser along CD is 

V Va 

Pj)V D log e — ^ • The work done is the difference of these two quantities, 

* D 

and the efficiency is 



P A V A \o<y e ^-P D V D \o» e ^ 
P A V A log e ^ 

V i 



T-t 



that of the Carnot cycle. Losses through the regenerator and by imper- 
fection of cycle reduce this in prac- 
tice. 



266. Entropy Diagram. This is 
given in Fig. 108. T and t are the 
limiting isothermals, DA and BO 
the constant volume curves, along 
each of which the increase of en- 
tropy is n = I log e ( T-f- 1), I being the 
specific heat at constant volume. 
The gain of entropy along the iso- 
thermals is obtained as in Art. 253. Ignoring the heat areas EDAF and 
GCBII, the efficiency is ABOD -h FABH, that of the Carnot cycle. The 
Stirling cycle appears in the PV diagram of Fig. 104 as dkbl. 



E F G H 

Fig. 103. Art. 266. — Stirling Cycle, 
Entropy Diagram. 



140 



APPLIED THERMODYNAMICS 



267. Importance of the Regenerator. Without the regenerator, the non- 
reversible Stirling cycle would have an efficiency of 



(^-^)F A log e 



K 



(T-t)+P A V A \og e 



V A 



This is readily computed to be far below that of the corresponding Carnot 
cycle. The advantage of the regenerative cycle lies in the utilization of 
the heat rejected along BC, Fig. 106, thus cancelling that item in the 
analysis of the cycle. Another way of utilizing this heat is to be 
described ; but while practical difficulties, probably insurmountable, limit 
progress in the application of the air engine on a commercial scale, the 
regenerator, upon which has been founded our modern metallurgical in- 
dustries as well, has offered the first possible method for the realization 
of the ideal efficiency of Carnot (2). 

268. Trials. As early as 1847, a 50-hp. Stirling engine, tested at the Dun- 
dee Foundries, was shown to operate at a thermal efficiency of 30 per cent, esti- 
mated to be equivalent, considering the rather low furnace efficiency, to a coal con- 
sumption of 1.7 lb. per hp.-hr. This latter result is not often surpassed by the aver- 
age steam engines of the present day. The friction losses in the mechanism were 
only 11 per cent (3). A test quoted by Peabody (4) gives a coal rate of 1.66 lb., 
but with a friction loss much greater, — about 30 per cent. There is no question 
as to the high efficiency of the regenerative air engine. 

269. Ericsson's Hot-air Engine. In 1833, Ericsson constructed an unsuccess- 
ful hot-air engine in London. About 1855, he built the steamer Ericsson, of 2200 
tons, driven by four immense hot-air engines. After the abandonment of this 
experiment, the same designer in 1875 introduced a third type of engine, and more 
recently still, a small pumping engine, which has been extensively applied. 

The principle of the engine of 
1855 is illustrated in Fig. 109. B is 
the receiver, A the displacer, II the 
furnace. The displacer A fits loosely 
in B excepting near its upper portion, 
where tight contact is insured by 
means of packing rings. The lower 
portion of A is hollow, and filled 
with a non-conductor. The holes 
aa admit air to the upper surface 
of A. D is the compressing pump, 
with piston C, which is connected 
with A by the rods dd. E is a pis- 
ton rod through which the de- 
veloped power is externally applied. Air enters the space above C through 
the check valve c, and is compressed during the up stroke into the magazine F 




Fig. 109. Arts. 209, 270, 275.— Ericsson Engine. 



ERICSSON ENGINE 



141 



through the second check valve e. G is the regenerator, made up of wire gauze. 
The control valves, worked from the engine mechanism, are at b and /. When 
b is opened, air passes from F through G to B, raising A. Closing of b at part 
completion of the stroke causes the air to work expansively for the remainder of 
the stroke. During the return stroke of A, air passes through G, f, and g to the 
atmosphere. 

270. Graphical Illustration. The PV diagram is given in Fig. 110. EBCF 
is the network diagram, ABCD being the diagram of the engine cylinder, AEFD 
that of the pump cylinder. Beginning with A in its lowest position, the state point 
in Fig. 110 is, for the engine (lower side of A), at 
A, and for the pump (upper side of C), at F. 
During about half the up stroke, the path in the 
engine is A B, air passing to B from the re- 
generator through s, and being kept at constant 
pressure by the heat from the furnace. During 
the second half of this stroke, the supply of air 
from the regenerator ceases, and the pressure falls 
rapidly as expansion occurs, but the heat im- 
parted from the furnace keeps the temperature 
practically constant, giving the isothermal path 
BC. Meanwhile, the pump, receiving air at the 

pressure of the atmosphere, has been first compressing it isothermally, or as 
nearly so as the limited amount of cooling surface will permit, along FE, and 
then discharging it through e at constant pressure, along EA, to the receiver F. 
On the down stroke, the engine steadily expels the air, now expanded down to 
atmospheric pressure, along the constant pressure line CD, while the pump simi- 
larly draws in air from the atmosphere at constant pressure along DF. At the end 
of this stroke, the air in F, at the state A, is admitted to the engine. The ratio of 

pump volume to engine volume is FD -±- DC, or — • 




Fig. 110. Arts. 270, 272, 273.— 
Ericsson Cycle. 




T 



Fig. 111. Art. 271.— Ericsson Cycle, 
Entropy Diagram. 



271. Efficiency. The Ericsson cycle be- 
longs to the same class as that of Stirling, 
being bounded by two isothermals and two 
like polytropics ; but the polytropics are in 
this case constant pressure lines instead of 
constant volume lines. The net entropy 
diagram EBCF, Fig. Ill, is similar to that 
of the Stirling engine, but the isodiabatics 
swerve more to the right, since k exceeds I, 

T-t 



while the efficiency is the same as that of the Stirling engine 



272. Tests. As computed by Eankine from Norton's tests, the effi- 
ciency of the steamer Ericsson's engines was 26.3 per cent ; the efficiency 
of the furnace was, however, only 40 per cent. The average effective pres- 



142 APPLIED THERMODYNAMICS 

sure (EBCF-t-XC, Fig. 110) was only 2.12 lb. The friction losses were 
enormous. A small engine of this type tested by the writer gave a con- 
sumption of 15.64 cu. ft. of gas (652 B. t. u. per cubic foot) per Ihp.-hr. ; 
equivalent to 170 B. t. u. per Ihp.-minute; and since 1 horse power 
= 33,000 foot-pounds =33,000 -§-778= 42.45 B. t. u. per minute, the 
thermodynamic efficiency of the engine was 42.45 -s- 170 = 0.25. 

273. Actual Designs. In order that the lines FC and EB, Fig. 110, may be 
horizontal, the engine should be triple or quadruple, as in the steamer Ericsson, in 
which each of the four cylinders had its own compressing pump, but all were con- 
nected with the same receiver, and with a single crank shaft at intervals of a 
quarter of a revolution. Specimen indicator diagrams are given in Figs. 107, 112. 

P 




Fig. 107. Art. 273. — Indicator Fig. 112. Art, 273. — Indicator 

Card from Ericsson Engine. Diagram, Ericsson Engine. 

274. Testing Hot-air Engines. It is difficult to directly and accurately meas- 
ure the limiting temperatures in an air engine test, so that a comparison of the 
actually attained with the computed ideal efficiencies cannot ordinarily be made. 
Actual tests involve the measurement of the fuel supplied, determination of its 
heating value, and of the indicated and effective horse power of the engine 
(Art. 487). These data permit of computation of the thermal and mechanical 
efficiencies, the latter being of much importance. In small units, it is sometimes 
as low as 0.50. 

275. The Air Engine as a Heat Motor. In nearly every large application, the 
hot-air engine has been abandoned on account of the rapid burning out of the 
heating surfaces due to their necessarily high temperature. Napier and Rankine 
(5) proposed an "air heater," designed to increase the transmissive efficiency of 
the heating surface. Modern forms of the Stirling or Ericsson engines, in small 
units, are comparatively free from this ground of objection. Their design permits 
of such amounts of heat-transmitting surface as to give grounds for expecting a 
much less rapid destruction of these parts. It has been suggested that excessive 
bulk may be overcome by using higher pressures. (Zeuner remarks (6) that the 
bulk is not excessive when compared with that of a steam engine with its auxiliary 
boiler and furnace). Rankine has suggested the introduction of a second com- 
pressed air receiver, in Fig. 109, from which the supply of air would be drawn 
through c, and to which air would be discharged through/. This would make the 
engine a " closed " engine, in which the minimum pressure could be kept fairly 
high; a small air pump would be required to compensate for leakage. A "con- 
denser " would be needed to supplement the action of the regenerator by more 



HOT-AIR ENGINES 143 

thoroughly cooling the discharged air, else the introduction of " back pressure " 
would reduce the working range of temperatures. The loss of the air by leakage, 
and consequent waste of power, would of course increase with increasing pressures. 
Instead of applying heat externally, as proposed by Joule, in the engine shown 
in Fig. 101, there is no reason why the combustion of the fuel might not proceed 
within the hot chamber itself, the necessary air for combustion being supplied by 
the pump. The difficulties arising from the slow transmission of heat would thus 
be avoided. An early example of such an engine applied in actual practice was 
Cayley's (7), later revived by Wenham (8) and Buckett (9). In such engines, 
the working fluid, upon the completion of its cycle, is discharged to the atmos- 
phere. The lower limit of pressure is therefore somewhat high, and for efficiency 
the necessary wide range of temperatures involves a high initial pressure in the 
cylinder. The internal combustion air engine even in these crude forms may be 
regarded as the forerunner of the modern gas engine. 

(1) Zeuner, Technical Thermodynamics (Klein), 1907, I, 340. (2) The theoreti- 
cal basis of regenerator design appears to have been treated solely by Zeuner, op. cit,, 
1,314-323. (3) Rankine, The Steam Engine, 1897, 368. (4) Thermodynamics of the 
Steam Engine, 1907, 302. (5) The Steam Engine, 1897, 370. (0) Op. cit., I, 381. 
(7) Nicholson's Art Journal, 1807; Min. Proc. Inst. C. E., IX. (8) Troc. Inst. 
Mech. Eng., 1873. (9) Inst. Civ. Eng., Heat Lectures, 1883-1884 ; Min. Proc. Inst. 
C. E., 1845, 1854. 



SYNOPSIS OF CHAPTER X 

The hot-air engine proper is an external combustion motor of the open or closed type. 
The temperature of a permanent gas may be varied independently of the pressure ; this 
makes the possible efficiency higher than that attainable in vapor engines. 

P\ / T\ 3-486 

—]=(—] ; the Carnot cycle leads to either excessive pressures or an enormous 
PI \tj 
cylinder. 
The polytropic cycle is bounded by two pairs of isocliabatics. 

The Lorenz cycle is bounded by a pair of adiabatics and a pair of unlike polytropics. 
The Beitlinger (isodiabatic) cycle is bounded by a pair of isothermals and a pair of 

isodiabatics. 
The Joule engine works in a cycle bounded by two constant pressure lines and two 

r r + 

adiabatics ; its efficiency is — . 

T 

The regenerator is a "fly wheel for heat." Any cycle bounded by a pair of iso- 
thermals and a pair of like polytropics, if worked with a regenerator, has an ideal 
efficiency equal to that of the Carnot cycle ; the heat rejected along one polytropic 
is absorbed by the regenerator, which in turn emits it along the other polytropic, 
the operation being subject to slight losses in practice. 

The Stirling cycle, bounded by a pair of isothermals and a pair of constant volume 
curves : correction of the ideal PV diagram for cushion air : comparison with indi- 
cator card ; the entropy diagram ; efficiency formulas with and without the regen- 
erator ; coal consumption, 1.7 lb. per hp.-hr. 

The Ericsson cycle, bounded by a pair of isothermals and a pair of constant pressure 
curves : efficiency from fuel to power, 26 per cent. 



144 APPLIED THERMODYNAMICS 

By designing as " closed' ' engines, the minimum pressure may be raised and the 

capacity of the cylinder increased. 
The air engine is unsatisfactory in large sizes on account of the rapid burning out of 

the heating surfaces and the small capacity for a given bulk. 



PROBLEMS 

(Note. Considerable accuracy in computation will be found necessary in solving Prob- 
lems 4 a and 5). 

1. How much greater is the ideal efficiency of an air engine working between tem- 
perature limits of 2900° F. and 600° F. than that of the steam engine described in Prob- 
lem 5, Chapter VI ? 

2. Plot to scale (1 inch = 2 cu. ft. = 40 lb. per square inch) the PV Carnot cycle 
for T=600°, £=500° (both absolute) the lowest pressure being 14.7 lb. per square 
inch, the substance being one pound of air, and the volume ratio during isothermal 
expansion being 12.6. 

3. In Problem 2, if the upper isothermal be made 700° absolute, what will be the 
maximum pressure ? 

4 a. Plot the entropy diagram, and find the efficiency, of a polytropic cycle for air 
between 600° F. and 500° F., in which n = 1.3, m = - 1.3, the pressure at d (Fig. 95) 
is 18 lb. per square inch, and the pressure at e (Fig. 95) is 22 lb. per square inch. 

4 6. In Art. 251, prove that 7> : T b : : T d : T e , and also that P d :P e ::P f : P b . 

5. Plot the entropy diagram, and find the efficiency, of a Lorenz cycle for air 
between 600° F. and 500° F., in which n = — 1.3, q = 0.4, the highest pressure being 
60 lb. per square inch and the temperature at g, Fig. 97, being 550° F. 

6. Plot the entropy diagram, and find the efficiency, of a Reitlinger cycle between 
600° F. and 500° F., when n = 1.3, the maximum pressure is 80 lb. per square inch, the 
ratio of volumes during isothermal expansion 12, and the working substance one 

pound of air. 

T— T 

7. Show that in the Joule engine the efficiency is 2 , Art. 255. 

8. Plot the entropy diagram, and find the efficiency, of a Joule air engine working 
between 600° F. and"— 200° F., the maximum pressure being 100 lb. per square inch, 
the ratio of volumes during adiabatic expansion 2, and the weight of substance 2 lb. 

9. Plot PV and NT diagrams for one pound of air worked between 3000° F. and 
400° F. : (a) in the Carnot cycle, (&) in the Ericsson cycle, (c) in the Stirling cycle, the 
extreme pressure range being from 50 to 2000 lb. per square inch. 

10. Find the efficiencies of the various cycles in Problem 9, without regenerators. 

11. Compare the efficiencies in Problems 4 a, 5, and 6, with that of the correspond- 
ing Carnot cycle. 

12. An air engine cylinder working in the Stirling cycle between 1000° F. and 
2000° F., with a regenerator, has a volume of 1 cu. ft. The ratio of expansion is 3. 
By what percentages will the capacity and efficiency be affected if the lower limit of 
pressure is raised from 14.7 to 85 lb. per square inch ? 

13. In the preceding problem, one eighth of the cylinder contents is cushion air, at 
1000° F. Plot the ideal indicator diagram for the lower of the two pressure limits, cor- 
rected for cushion air. 



HOT-AIR ENGINES 145 

14. In Art. 268, assuming that the coal used in the Dundee foundries contained 
14,000 B. t. u. per pound, what was the probable furnace efficiency? In the Peabody 
test, if the furnace efficiency was 80 per cent, and the coal contained 14,000 B. t. u., 
what was the thermal efficiency of the engine ? 

15. What was the efficiency of the plant in the steamer Ericsson ? 

16. Sketch the TiVand PV diagrams, within the same temperature and entropy 
limits, of all of the cycles discussed in this chapter, with the exception of that of Joule. 
Why cannot the Joule and Ericsson cycles be drawn between the same limits? Show 
graphically that in no case does the efficiency equal that of the Carnot cycle. 

17. Compare the cycle areas in Problem 9. 

18. In Problem 2, what is the minimum possible range of pressures compatible 
with a finite work area ? Illustrate graphically. 

19. Derive a definite formula for the efficiency of the Reitlinger cycle, Art. 253. 



CHAPTER XI 

GAS POWER 

The Gas Producer 

276. History. The bibliography (1) of internal combustion engines is exten- 
sive, although their commercial development is of recent date. Coal gas was dis- 
tilled as early as 1691 ; the waste gases from blast furnaces were first used for 
heating in 1809. The first English patent for a gas engine approaching modern 
form was granted in 1791. The advantage of compression was suggested as early 
as 1801, but was not made the subject of patent until 1838 in England and 1861 in 
France. Lenoir, in 1860, built the first practical gas engine, which developed a 
thermal efficiency of 0.04. The now familiar polytropic " Otto " cycle was pro- 
posed by Beau de Rochas at about this date. The same inventor called attention 
to the necessity of high compression pressures in 1862 ; a principle applied in 
practice by Otto in 1871. Meanwhile, in 1870, the first oil engine had been built. 
The four-cycle compressive Otto " silent " engine was brought out in 1876, show- 
ing a thermal efficiency of 0.15, a result better than that then obtained in the best 
steam power plants. 

If the isothermal, isometric, isopiestic, and adiabatic paths alone are considered, 
there are possible at least twenty-six different gas engine cycles (2). Only four 
of these have had extended development ; of these four, only two have survived. 
The Lenoir (3) and Hugon (4) non-compressive engines are now represented only 
by the Bischoff (5). The Barsanti "free piston" engine, although copied by 
Gilles and by Otto and Langen (1866) (6), is wholly obsolete. The variable vol- 
ume engine of Atkinson (7) was commercially unsuccessful. 

Up to 1885, illuminating gas was commonly employed, only small engines 
were constructed, and the high cost of the gas prevented them from being com- 
mercially economical. Nevertheless, six forms were exhibited in 1887. The 
Priestman oil engine was built in 1888. With the advent of the Dowson process, 
in 1878, with its possibilities of cheap gas, advancement became rapid. By 1897, 
a 400-hp. four-cylinder engine was in use on gas made from anthracite coal. At 
the present time, double-acting engines of 5400 hp. have been placed in operation ; 
still larger units have been designed, and a few applications of gas power have 
been made even in marine service. 

Natural gas is now transmitted to a distance of 200 miles, under 300 lb. pres- 
sure. Illuminating gas has been pumped 52 miles. Martin (8) has computed that 
coal gas might be transmitted from the British coal fields to London at a delivered 
cost of 15 cents per 1000 cu. ft. His plan calls for a 25-inch pipe line, at 500 lb. 
initial pressure and 250 lb. terminal pressure, carrying 40,000,000,000 cu. ft. of 

146 



GAS POWER 147 

gas per year. The estimated 46,000 hp. required for compression would be derived 
from the waste heat of the gas leaving the retorts. 

Producer gas is even more applicable to heating operations than for power 
production. It is meeting with extended use in ceramic kilns and for ore roast- 
ing, and occasionally even for firing steam boilers. 

277. The Gas Engine Method. The expression for ideal efficiency, 
(T— t)-r-T, increases as T increases. In a steam plant, although boiler fur- 
nace temperatures of 2500° F. or higher are common, the steam passes to 
the engine, ordinarily, at not over 350° F. This temperature expressed in 
absolute degrees limits steam engine efficiency. To increase the value of 
T, either very high pressure or superheat is necessary, and the practicable 
amount of increase is limited by considerations of mechanical fitness to 
withstand the imposed pressures or temperatures. In the internal com- 
bustion engine, the working substance reaches a temperature approximat- 
ing 3000° F. in the cylinder. The gas engine has therefore the same ad- 
vantage as the hot air engine, — a wide range of temperature. Its working 
substance is, in fact, for the most part heated air. The fuel, which may 
be gaseous, liquid, or even solid, is injected with a proper amount of air, 
and combustion occurs within the cylinder. The disadvantage of the ordi- 
nary hot air engine has been shown to arise from the difficulty of trans- 
mitting heat from the furnace to the working substance. In this respect, 
the gas engine has the same advantage as the steam engine, — large capa- 
city for its bulk, — for there is no transmission of heat ; the cylinder is 
the furnace, and the products of combustion constitute the working sub- 
stance. A high temperature of working substance is thus possible, with 
large work areas on the pv diagram, and a rapid rate of heat propagation. 

In the gas engine, then, certain chemical changes which constitute the pro- 
cess described as combustion, must be considered ; although such changes are in gen- 
eral not to be included in the phenomena of engineering thermodynamics. 

278. Fuels. The common fuels are gases or oils. In some sections, natural 
gas is available. This is high in heating value, consisting mainly of methane, 
CH 4 . Carbureted water gas, used for illumination, is nearly as high in heating 
value, consisting of approximately equal volumes of hydrogen, carbon monoxide, 
and methane, with some ethylene and traces of other substances. Uncarbureted 
(blue) water gas is almost wholly carbon monoxide and hydrogen. Its heating 
value is less than half that of the carbureted gas. Both water gas and coal gas 
are uneconomical for power production ; in the processes of manufacture, large 
quantities of coal are left behind as coke. Coal gas, consisting principally of hy- 
drogen and methane, is slightly lower in heating value than carbureted water 
gas. It is made by distilling soft coal in retorts, about two thirds of the weight 
of coal becoming coke. Coke oven gas is practically the same product; the main 
output in its case being coke, while in the former it is gas. 



148 APPLIED THERMODYNAMICS 

Producer gas ("Dowson" gas, "Mond" gas, etc.) is formed by the par- 
tial combustion of coal in air. It is essentially carbon monoxide, diluted 
with large quantities of nitrogen and consequently low in heating value. 
Its exact composition varies according to the fuel from which it is made, 
the quantity of air supplied, etc. When soft coal is used, or when much 
steam is fed to the producer, large proportions of hydrogen are present. 

It is of no value as an ilium inant. Blast furnace gas is producer gas 
obtained as a by-product on a large scale in metallurgical operations. It contains 
less hydrogen than ordinary producer gases, since steam is not employed in its 
manufacture, and is generally quite variable in its composition on account of the 
exigencies of furnace operation. Acetylene, C 2 H 2 , is made by combining calcium 
carbide and water. It has an extremely high heating and illuminating value. 
All hydrocarbonaceous substances maybe gasified by heating in closed vessels; 
gases have in this way been produced from peat, sawdust, tan bark, wood, garbage, 
animal fats, etc. 

279. Oil Gases. Many liquid hydrocarbons may be vaporized by appropriate- 
methods, under conditions which make them available for gas engine use. Some 
of these liquids must be vaporized by artificial heat and then immediately used, or 
they will again liquefy as their temperatures fall. The vaporizer or u carburetor " 
is therefore located at the engine, where it atomizes each charge of fuel as required. 
Gasoline is most commonly used ; its vapor has a high heating value. Kerosene, 
and, more recently, alcohol, have been employed. By mixing gasoline and air in 
suitable proportions, a saturated or " carbureted " air is produced. This acts as 
a true gas, and must be mixed with more air to permit of combustion. A gas 
formed in the proportion of 1000 cu. ft. of air to 2 gallons of liquid gasoline, for 
example, does not liquefy. A third form of oil gas is produced by heating certain 
hydrocarbons without air; the "cracking" process produces, first, less dense 
liquids, and, finally, gaseous bodies, which do not condense. The process must be 
carried on in a closed retort, and arrangements must be made for the removal of 
residual tar and coke. 

280. Liquid Fuels. These have advantages over solid or gaseous fuels, aris- 
ing from the usually large heating value per unit of bulk, and from ease of trans- 
portation. All animal and vegetable oils and fats may be reduced to liquid fuels; 
those oils most commonly employed, however, are petroleum products. Crude 
petroleum may be used; it is more customary to transform this to "fuel oil" by 
removing the moisture, sulphur, and sediment; and some of these "fuel oils" are 
used in gas engines. Of petroleum distillates, the gasolines are most commonly 
utilized in this country. They include an 86° liquid, too dangerous for commer- 
cial purposes; the 74° "benzine," and the 69° naphtha. "Distillate," an impure 
kerosene, from which the gasoline has not been removed, is occasionally used. 
Both grain alcohol (C 2 H 6 0) and wood alcohol (CH 4 0) have been used in gas en- 
gines (9). Various distillates from brown and hard coal tars have been employed 
in Germany. Their suitability for power purposes varies with different types of 
engines. The benzol derived from coal gas tar has been successfully used ; the 
brown coal series, C n H 2 „, C n H 2n +2, C n H 2n _ 2 , contains many useful members (10). 



THE GAS PRODUCER 



149 



281. The Gas Producer. This essential auxiliary of the modern gas 
engine is made in a large number of types, one of which is shown in Fig. 
113. This is a brick-lined cylindrical shell, set over a water-sealed pit P, 
on which the ash bed rests. Air is forced in by means of the steam jet 
blower A, being distributed by means of the conical hood B, from which 




Fig. 113. Art. 281. — The Amsler Gas Producer. 



it passes up to the red-hot coal bed above. Here carbon dioxide is formed 
and the steam decomposes into hydrogen and oxygen. Above this " com- 
bustion zone " extends a layer of coal less highly heated. The carbon 
dioxide, passing upward, is decomposed to carbon monoxide and oxygen. 
The hot mixed gases now pass through the freshly fired coal at the top of 
the producer, causing the volatile hydrocarbons to distill off, the entire 
product passing out at C. The coal is fed in through the sealed hopper D. 



150 APPLIED THERMODYNAMICS 

At E are openings for the bars used to agitate the fire. At F are peep- 
holes. 

An automatic feeding device is sometimes used at D. The air may 
be forced in by a blower, or sucked through by an exhauster, or by the 
engine piston itself, displacing the steam jet blower A. The fuel may 
be supported on a solid grate, or on the bottom of a producer without the 
water seal; grates may be either stationary or mechanically operated. 
Mechanical agitation may be employed instead of the poker bars inserted 
through E. Sometimes water gas, for illumination, and producer gas, for 
power, are made in the same plaut. Two producers are then employed, 
the air blast being applied to one, while steam is decomposed in the other. 

Provision must be made for purifying the gas, by deflectors, wet and dry 
scrubbers, filters, coolers, etc. For the removal of tar, which would be seriously 
objectionable in engines, mechanical separation and washing are useful, but the 
complete destruction of this substance involves the passing of the gas through a 
highly heated chamber; this may be a portion of the producer itself, as in 
"under-feed," "inverted combustion," or "down-draft" types: causing the trans- 
formation of the tar to fixed gases. On account of the difficulty of tar removal, 
anthracite coal or coke or semi-bituminous, non-caking coal must generally be used 
in power plants. The air supplied to the producer is sometimes preheated by the 
sensible heat of the waste gases, in a " recuperator." The " regenerative " prin- 
ciple — heating the air and gas delivered to the engine by means of the heat of 
the exhaust gases — is inapplicable, for reasons which will appear. 

282. The Producer Plant. The ordinary producer operates under a slight 
pressure ; in the suction type, now common in small plants, the engine piston 
draws air through the producer in accordance with the load requirements. Pres- 
sure producers have been used on extremely low grade fuels: Jahn, in Germany, 
has, it is reported, gasified mine waste containing only 20 per cent of coal. Suc- 
tion producers, requiring much less care and attention, are usually employed only 
on the better grades of fuel. Most producers require a steam blast ; the steam 
must be supplied by a boiler or " vaporizer," which in many instances is built as a 
part of the producer, the superheated steam being generated by the sensible heat 
carried away in the gas. Automatic operation is effected in various ways : in 
the Amsler system, by changing the proportion of hydrogen in the gas, involving 
control of the steam supply ; in the Pintsch process, by varying the draft at the 
producer by means of an inverted bell, under the control of a spring, from beneath 
which the engine draws its supply; and in the Wile apparatus, by varying the 
draft by means of valves operated from the holder. Figure 114 shows a complete 
producer plant, with separate vaporizer, economizer (recuperator), and holder for 
storing the gas and equalizing the pressure. 

283. By-product Recovery. Coal contains from 0.5 to 3 per cent of nitrogen, 
about 15 per cent of which passes off in the gas as ammonia. The successful 
development of the Mond process has demonstrated the possibility of recovering 
this in the form of ammonium sulphate, a valuable fertilizing agent. 



THE GAS PRODUCER 



151 




152 APPLIED THERMODYNAMICS 

284. Action in the Producer. Coal is gasified on the producer grate. 
Ideally, this coal is carbon, and leaves the producer as carbon monoxide, 
4450 B. t. u. per pound of carbon having been expended in gasification. 
Then only 10,050 B. t. u. per pound of carbon are present in the gas, and 
the efficiency cannot exceed 10,050 -r- 14,500 = 0.694. The 4450 B. t. u. con- 
sumed in gasification are evidenced only in the temperature of the gas. 
With actual conditions, the presence of carbon dioxide or of free oxygen 
is an evidence of improper operation, further decreasing the efficiency. By 
introducing steam, however, decomposition occurs in the producer, the tem- 
perature of the gas is reduced, and available hydrogen is carried to the 
engine ; and this action is essential to producer efficiency for power pur- 
poses, since a high temperature of inlet gas is a detriment rather than a 
benefit in engine operation. The ideal efficiency of the producer may thus 
be brought up to something over 80 per cent; a limit arising when the 
proportion of steam introduced is such as to reduce the temperature of the 
gas below about 1800° F., when the rate of decomposition greatly decreases. 
The proportion of steam to air, by weight, is then about 6 per cent, the 
heating value of the gas is increased, the percentage of nitrogen decreased, 
and nearly 20 per cent of the total oxygen delivered to the producer has 
been supplied by decomposed steam. A similar result may be attained by 
introducing exhausted gas from the engine to the producer. The carbon 
dioxide in this gas decomposes to monoxide, which is carried to the engine 
for further use. This method is practiced in the Mond system, and has 
had other applications. To such extent as the coal is hydrocarbonaceous, 
however, the ideal efficiency, irrespective of the use of either steam or 
waste gas, is 100 per cent. Figure 115 shows graimically the results com- 
puted as following the use of either steam or waste gases with pure car- 
bon as the fuel. The maximum ideal efficiency is about 3J per cent greater 
when steam is used, if the temperature limit is fixed at 1800° F., but the 
waste gases give a more uniform (though less rich) gas. The higher ini- 
tial temperature of the waste gases puts their use practically on a parity 
with that of steam. Either system tends to prevent clinkering. The 
maximum of producer efficiency, for power gas purposes, is ideally from 
5 to 10 per cent less than that of the steam boiler. High percentages of 
hydrogen resulting from the excessive use of steam may render the gas 
too explosive for safe use in an engine (10 a) (25). 

285. Example of Computation. Let 20 per cent of the oxygen necessary for 
gasifying pure carbon be supplied by steam. Each pound of fuel requires \\ lb. 
of oxygen for conversion to carbon monoxide. Of this amount, 0.20 x 1^ = 0.2666 lb. 
will then be supplied by steam ; and the balance, 1.0667 lb., will be derived from 
the air, bringing in with it Jf x 1.0667 = 3.57 lb. of nitrogen. The oxygen derived 
from steam will also carry with it \ x 0.2666 = 0.0333 lb. of hydrogen. The pro- 
duced gas will contain, per pound of carbon, 



PRODUCER EFFICIENCY 



153 



2.33 lb. carbon monoxide, 
3.57 lb. nitrogen, 
0.0333 lb. hydrogen. 

The heat evolved in burning to monoxide is 4450 B. t. u. per pound. A por- 
tion of this, however, has been put back into the gas, the temperature having been 



Waste Gas supplied; Percentage of Fuel gasified by Weight 
109 202 256 382 



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o.y 




<y 






0- 




o 


J 


0./ 


C 


rT" 




X 


"S" 


0.5 








+- 


.y 




r 


it 




Q> 


LU 

-o>- 


03 


o 

i- 
a 


E 






-O- 


0.1 




£ 







lowered by the decomposition of the steam. Under the conditions existing in the 
producer, the heat of decomposition is about 62,000 B. t. u. per pound of hydrogen. 
The net amount of heat evolved is then 4450 - (0.0333 x 62,000) = 2383 B. t. u., 



154 APPLIED THERMODYNAMICS 

14 500 9 C 5S C 5 

and the efficiency is — - — = 0.84. The rise in temperature is computed as 

follows : to heat the gas 1° F. there are required 

Weight Specific Heat 

For carbon monoxide, 2.33 x 0.2479 = 0.578 B. t. u. 

For nitrogen, 3.57 X 0.2438 = 0.869 B. t. u. 

For hydrogen, 0.0333 x 3.4 = 0.113 B. t. u. 

a total of 1.560 B. t. u. 

The 2383 B. t. u. evolved will then cause an elevation of temperature of 

-^ = 1527° F. 
1.560 

With pure air only, used for gasifying pure carbon, the gas w^ould consist of 
2\ lb. of carbon monoxide and 4.45 lb. of nitrogen; the percentages being 34.5 
and 65.5. For an actual coal, the ideal gas composition may be calculated on the 
assumptions that the hydrogen and hydrocarbons pass off unchanged, and that the 
carbon requires 1| times its own weight of oxygen, part of which is contained in 
the fuel, and part derived from steam or from the atmosphere, carrying with it 
hydrogen or nitrogen. Multiplying the weight of each constituent gas in a pound 
by its calorific value, we have the heating value of the gas. As a mean of 54 
analyses, Fernald finds (11) the following percentages by volume : 

Carbon monoxide (CO) 19.2 

Carbon dioxide (CO->) 9.5 

Hydrogen (H) 12.4 

Marsh gas and ethylene (CH 4 , C 2 H 4 ) 3.1 

Nitrogen (H) 55.8 

100.0 

286. Figure of Merit. A direct and accurate determination of efficiency is 
generally impossible, on account of the difficulties in gas measurement (12). For 
comparison of results obtained from the same coals, the figure of merit is sometimes 
used. This is the quotient of the heating value per pound of the gas by the 
weight of carbon in a pound of gas : it is the heating value of the gas per pound of 
carbon contained. In the ideal case, for pure carbon, its value would be 10,050 B. t. u. 
For a hydrocarbonaceous coal, it may have a greater value. 



Gas Engine Cycles 

287. Four-cycle Engine. A gas engine of one of the most commonly used 
types is shown in Fig. 116. This represents a single-acting engine; i.e. the gas is 
in contact with one side of the piston only, the other end being open. Large en- 
gines of this type are frequently made double-acting, the gas being then con- 
tained on both sides of a piston moving in an entirely closed cylinder, exhaust 
occurring on one side while some other phase of the cycle is described on the 
other side. 



THE GAS ENGINE 



155 




Fig. 116. Art. 287. — Single-acting Gas Engine, Four Cycle. 
(From " The Gas Engine," by Cecil P. Poole, with the permission of the Hill Publishing Company.) 




Fig. 117. Art. 288.— Piston Movements, Otto Cycle. 
(From "The Gas Engine," by Cecil P. Poole, with the permission of the Hill Publishing Company.) 



156 APPLIED THERMODYNAMICS 

288. The Otto Cycle. Figure 117 illustrates the piston move- 
ments corresponding to the ideal pv diagram of Fig. 118. The 
cycle includes five distinctly marked paths. During the out stroke 
of the piston from position A to position B, Fig. 117, gas is sucked 

in by its movement, giving the line 
ab, Fig. 118. During the next in- 
ward stroke, B to (7, the gas is com- 
pressed, the valves being closed, 
along the line be. The cycle is not 
yet completed : two more strokes 
are necessary. At the beginning 




Italia Arts. 288, 291.-The Otto Cycle. of ^ firgt rf ^^ ^ pigton 

being at <?, Fig. 118, the gas is ignited and practically instantaneous 
combustion occurs at constant volume, giving the line eO. An out 
stroke is produced, and as the valves remain closed, the gas expands, 
doing work along Cd, while the piston moves from C to 2), Fig. 117. 
At c?, the exhaust valve opens, and during the fourth stroke the 
piston moves in from D to E, expelling the gas from the cylinder 
along de, Fig. 118. This completes the cycle. The inlet valve has 
been open from a to S, the exhaust valve from d to e. During the 
remainder of the stroke, the cylinder was closed. Of the four 
strokes, only one was a " working " stroke, in which a useful effort 
was made upon the piston. In a double-acting engine of this type, 
there would be two working strokes in every four. 

289. Two-stroke Cycle. Another largely used type of engine is shown 
in Fig. 119. The same five paths compose the cycle ; but the events are 
now crowded into two strokes. The exhaust opening is at E ; no valve 
is necessary. The inlet valve is at A, and ports are provided at C, C and 
i". The gas is often delivered to the engine by a separate pump, at a 
pressure several pounds above that of the atmosphere ; in this engine, the 
otherwise idle side of a single-acting piston becomes itself a pump, as 
will appear. Starting in the position shown, let the piston move to the left. 
It draws a supply of combustible gas through A, B and the ports C into 
the chamber D. On the outward return stroke, the valve A closes, and the 
gas in D is compressed. Compression continues until the edge of the piston 
passes the port I, when this high pressure gas rushes into the space F, at 
practically constant pressure. The piston now repeats its first stroke. 
Following the mass of gas which we have been considering, we find that 



THE TWO-CYCLE ENGINE 



157 



it undergoes compression, beginning as soon as the piston closes the ports 
E and I, and continuing to the end of the stroke, when the piston is in its 
extreme left-hand position. Ignition there takes place, and the next out 




Fig. 119. Arts. 289-291, 309, 339.— Two-cycle Gas Engine. 
(From " The Gas Engine," by Cecil P. Poole, with the permission of the Hill Publishing Company.) 

stroke is a working stroke, during which the heated gas expands. Toward 
the end of this stroke, the exhaust port E is uncovered, and the gas passes 
out, and continues to pass out until early on the next backward stroke this 
port is again covered. 

290. Discussion of the Cycle. We have here a two-stroke cycle ; for 
two of the four events requiring a perceptible time interval are always 
taking place simultaneously. On the first stroke to the left, while gas is 
entering D, it is for a brief interval of time also flowing from i" to F, from 
F through E, and afterward being compressed in F. On the next stroke 
to the right, while gas is compressed in D, ignition and expansion occur in 
F; and toward the end of the stroke, the exhaust of the burned gases 
through E and the admission of a fresh supply through /, both begin. 
The inlet port / and the exhaust port E are both open at once during part 
of the operation. To prevent, as far as possible, the fresh gas from 
escaping directly to the exhaust, the baffle G is fixed on the piston. It is 
only by skillful proportioning of port areas, piston speed, and pressure in 
D that large loss from this cause is avoided. The burned gases in the 
cylinder, it is sometimes claimed, form a barrier between the fresh enter- 
ing gas and the exhaust port. 



158 



APPLIED THERMODYNAMICS 



291. PV Diagram. This is shown for the working side (space F) in 

Fig. 120 and for the pumping side (space D) in Fig. 121. The exhaust 

port is uncovered at d, Fig. 120, and the pres- 
sure rapidly falls. At a, the inlet port opens, 
the fresh supply of gas holding up the pres- 
sure. From a out to the end of the diagram, 
and back to b, both ports are open. At b the 
inlet port closes, and at c the exhaust port, 
when compres- 
sion begins. The 

v pump diagram of 

Two-stroke Fig. 121 corre- 
sponds with the 

negative loop deab of Fig. 118. Aside from 

the slight difference at dabc, Fig. 120, the 

diagrams for the two-cycle and four-cycle engines are precisely the same ; 

and in actual indicator cards, the difference is very slight. 




Fig. 120. Art. 291. 
Cycle 



Fig. 121. Art. 291.— Two-stroke 
Cycle Pump Diagram. 



292. Ideal Diagram. The perfect PV 
diagram for either engine would be that of 
Fig. 122, ebfd, in which expansion and com- 
pression are adiabatic, combustion instan- 
taneous, and exhaust and suction unre- 
Fig. 122. Arts. 292, 293, 294, dieted; so that the area of the negative 
295, 314, 329, 331, Prob. 15.— loop dg becomes zero, and eb and fd are 

Idealized Gas Engine Dia- lines of congtant vo l ume . From inspection 



p 


*'rv 








el 


^z. 








'^T^ 


/ 




(1 


^"^"--^^^ 


(f 









gram. 



of the diagram we find 



p e r e "=p d VA ■ 


*'-*<W 


) 


p,v/ = p b r b y, v r - 


= r d , 


'-*<©"• 


rp rp ■* b 

J-b— Je-^T, 

■*■ e 

T — T f 

-id 




*/r *<*)'. r.. 


= r* 


293. Work Done. The work area un 


i w P b V b -P f V f 

der of is b b f f ; 

y- 1 

of the cycle is 


that 


under ed is — - — — 

y 


P V 

d d ; the net 


work 






P*Vi+P d V 4 - 


P f V, 


- P V 

M e r e 






y- 


1 







EFFICIENCY OF OTTO CYCLE 



159 



This may be written in terms of two pressures and two volumes only, 
for P e V e = P d V d y V^-y and P f V f =P b V? V^, giving 



w= 



p b v b + P d v d - p b v b y v ( }-y - p d v/ v b 



\-y 



P h V b +V 



y 



y-i 



p d - 



p - 



v.v-y ^ " 

y b \ p 

-L d 



V d 



-JJ 



P fU \v p 

294. Relations of Curves. Expressing -^ = [ — f -\ and — ^ 



Pf 



, and 



This 



y 
p^ 

Pa P 6 Pa 

permits of rapidly plotting one of the curves when the other is given. 
T h T, , T h T, 



p p p 

remembering that V h = V e , V f = V d , we have ~ = ^ and ^=. 



We also find 



and |» = 



f 



295. Efficiency. In Fig. 122, heat is absorbed along eb, equal to 
l(T b — Tg); this is derived from the combustion of the gas. Heat 
is rejected along fd, — l{T f — T^). Using the difference of the two 
quantities as an expression for the work done, we obtain for the 
efficiency 
T b -T e -T f +T d 



= 1 + 


T d - 

T b - 




= 1- 


_T b - 


-Tf. 


T ■ 

— e 


-T d 



Ti 



T h 



T. 



P 



The efficiency thus depends solely upon the extent of compression. 

296. Carnot Cycle and Otto Cycle; the Atkinson Engine. Let abed, 
Fig. 123, represent a Carnot cycle drawn to pv coordinates, and bfde, the 
corresponding Otto cycle between the p 
same temperature limits, T and t. For the 
Carnot cycle, the efficiency is (T — t) -=- T; 
for the Otto, it is, as has been shown, 
(T e - T d ) -f- T e . It is one of the disad- 
vantages of the Otto cycle, as shown in 
Art. 294, that the range of temperatures 
during expansion is the same as that dur- 
ing compression. In the ingenious Atkin- 
son engine (13), the fluid was contained in 
the space between tivo pistons, which space 
was varied during the phases of the cycle. This permitted of expansion 
independent of compression; in the ideal case, expansion continued down 




Fig. 123. Art. 296.— Carnot, Otto, 
and Atkinson Cycles. 



160 



APPLIED THERMODYNAMICS 



to the temperature of the atmosphere, giving such a diagram as ebcd, Fig. 

123. The entropy diagrams for the Carnot, Otto, and Atkinson cycles are 

correspondingly lettered in Fig. 124. For 
the Atkinson cycle, in the ideal case, we 
have in Fig. 124 the elementary strip 
vivxy, which may stand for dH, and the 
isothermal dc at the temperature t. Let 
the variable temperature along eb be T x , 
having for its limits T b and T e . Then for 
the area ebcd, we have 




Fig. 124. Arts. 29G, 297, 305, 307.— 
Efficiencies of Gas Engine Cycles. 



= l{T b -T t ) 



It log, S- 



The efficiency is obtained by dividing by I (T b — T e ) and is equal to 



1- 



log e ^ 



297. Application to a Special Case. Let T e = 1060, T = 3440, t = 520 ; 
whence, from Art. 294, T f = 1688. We then have the following ideal efficiencies : 

T - t 3440 - 520 



Carnot, 



Atkinson, 1 



3440 



Tb 



t . T h 
ZTT^Te 



= 0.85. 

520 3440 _ 
2380 ge 1060" 



0.74. 



Otto, 



T e - t 1060 - 520 



= 0.51. 



T e 1060 

The Atkinson engine can scarcely be regarded as a practicable type ; the 
Otto cycle is that upon which most gas engine efficiencies must be based ; 
and they depend solely -on the ratio of temperatures or pressures during 
compression. 

298. Lenoir Cycle. This is shown in Fig. 125. The fluid is drawn 
into the cylinder along Ad and exploded along df. Expansion then 
occurs, giving the path^, when the exhaust valve opens, the pressure 





CONSTANT VOLUME 



Fig. 125. Arts. 298, 301, 302.— 
Lenoir Cycle, 



Fig. 126. Art. 298. — Entropy 
Diagram, Lenoir Cycle. 



BRAYTON CYCLE 



161 



falls, gh, until it reaches that of the atmosphere, and the gases are finally 
expelled on the return stroke, JiA. It is a tivo-cycle engine. The net 
entropy diagram appears in Fig. 126. 
The efficiency is 

Heat absorbed - heat rejected _ l(T f - T d ) - l(T g - T h ) - k(T n - T d ) 



Heat absorbed 



T 



KT f - T d ) 

T h T h - T d 
-Ffr-y-n 



lf-*d If — Id 

Brayton Cycle. This is shown in Fig. 127. A separate 
pump is employed. The substance is drawn in along Ad, compressed 
along dn, and forced into a reservoir along nB. The engine begins 
to take a charge from the reservoir at B, which is slowly fed in and 
ignited as it enters, so that combustion proceeds at the same rate as 
the piston movement, giving the constant pressure line Bb. Expan- 





Fig. 127. Arts. 299, 302.— Brayton 
Cycle. 



Fig. 128. Art. 299. — Brayton Cycle, 
Entropy Diagram. 



sion then occurs along bg, the exhaust valve opens at g, and the 
charge is expelled along hA. The net cycle is dnbgh; the net ideal 
entropy diagram is as in Fig. 128. This is also a two-cycle 
engine. The " constant pressure " cycle which it uses was suggested 
in 1865 by Wilcox. In 1873, when first introduced in the United 
States, it developed an efficiency of 2.7 lb. of (petroleum) oil per 
brake hp.-hr. 

The efficiency is (Fig. 127) 



1- 



T h 



T h -T, 



KT b -T n ) - KT q - T h )-k(T h -T d ) _ 

KT b -T n } '- y(T b -T n ) T b -T n 

If expansion is complete, the cycle becoming dnbi, Figs. 127, 128, then 
T g = T h = Ti, and the efficiency is 

rji ni rrt rp rjn 

1 _ ' ~ d — i zA — n ~ ± ^ 

T h -T n ~ T n ~ T n > 

a result identical with that in Art. 295 ; the efficiency (with complete ex- 
pansion) depends solely upori the extent of compression. 



162 APPLIED THERMODYNAMICS 

300. Comparisons with the Otto Cycle. It is proposed to compare the capacities 
and efficiencies of engines working in the Otto,* Brayton, and Lenoir cycles ; the 
engines being of the same size, and working between the same limits of temperature. 
For convenience, pure air will be regarded as the working substance. In each case 
let the stroke be 2 ft., the piston area 1 sq. ft., the external atmosphere at 17° C, 
the maximum temperature attained, 1537° C. In the Lenoir engine, let ignition 
occur at half stroke ; in the Brayton, let compression begin at half stroke and con- 
tinue until the pressure is the same as the maximum pressure attained in the Lenoir 
cycle, and let expansion also begin at half stroke. These are to be compared with 
an Otto engine, in which the pump compresses 1 cu. ft. of free air to 40 lb. net 
pressure. This quantity of free air, 1 cu. ft., is then supplied to each of the three 
engines. 

301. Lenoir Engine. The expenditure of heat (in work units) along df, Fig. 
125, is Jl(T — t), in which T = 1537, t = 17, J is the mechanical equivalent of a 
Centigrade heat unit, and I is the specific heat of 1 cu. ft. of free air, 
heated at constant volume 1° C. Now J = 778 xf- 1400.4, and J I is approxi- 
mately 0.1689 x 0.075 x 1400.4 = 17.72. The expenditure of heat is then 

17.72(1537 - 17) = 26,900 ft.-lb. 
The pressure at /is 

14.7 153T + 273 = 91.4 lb. absolute; 
17 + 273 

and the pressure at g is 

91.4 (f)* = 34.25 lb. absolute. 

The work done under fg is then 

144 { (9L4xl)-(34.25x2) J = gl90 ^^ 
1 1.402 - 1 J 

The negative work under hd is 14.7 x 144 x 1 = 2107 ft.-lb., and the net work is 
8190 - 2107 = 6083 ft.-lb. The efl&ciency is then 6083 -=- 26,900 = 0.226. 

302. Brayton Engine. We first find (Fig. 127) 

T n =T d (^A v = (273 + 17) (®±AY' m = 489° absolute or 216° C. 

Proceeding in the same way as with the Lenoir engine, we find the heat expendi- 
ture to be 

Jk(T h - T n )= 0.2375 x 0.075 x 1400.4(1537 - 216)= 33,000 ft.-lb. 

The pressure at n is by assumption equal to that in the case of the Lenoir engine ; 
the pressure at g in the Brayton type then equals that at g in the Lenoir. The 
work under bg is the same as that under fg in Fig. 125. The work under nb is 
found by first ascertaining the volume at n. This is 



fMIV^l.O =0.272. 
V9.14/ 



* The " Otto cycle " in this discussion is a modified form (as suggested by Clerk) 
in which the strokes are of unequal length. 



CLERK'S GAS ENGINE 163 

The work under nb is then 91.4 x 144 x (1.0 - 0.272) = 9650 ft.-lb., and the gross 
work is 9650 + 8190 = 17,840 ft.-lb. Deducting the negative work under hd, 
2107 ft.-lb., and that under dn, 

1U /(91.4 x 0.272) - (14.7 x 1)_N = ft> _ 

V 1.402 - 1.0 / 

the net work area is 12,083 ft.-lb., and the efficiency, 12,083 -f- 33,000 = 0.366. 

303. Clerk's Otto Engine. In Fig. 129, a separate pump takes in a charge 
along AB, and compresses it along BC, afterward forcing it into a receiver along 
CD at 40 lb. gauge pressure. Gas flows from p 
the receiver into the engine along DC, is ex- 
ploded along CE, expands to F, and is expelled 
along GA. The net cycle is BCEFG. The 
volume at C is 

(I^V + "= 0.393 cu. ft. 

\«l-<> Fig. 129. Arts. 303, 305.— Clerk's 

The temperature at C is otto °y cle - 

P.V.T^ p iF>= (54.7 x 6.393X273 + 17) _ 273 = 153 o c . 
14.7 x 1 
The pressure at E is then 

(15 ?l +2 L 3 f 4 - 7 = 231 lb. absolute. 
153 + 273 

The pressure at F is 

23l(M^?V = 23.64 lb. absolute. 
The work under EF is 

U4 / (231x0.393)-(23.64x2) \ = 15 




/ (231 x 0.393) - (23.64 x 2) \ 
V 1.402-1.0 I 



that under BG is 2107 ft.-lb., and that under BC is 

1M /(54.7x 0.393)- (14.7xm =2430 ft ._ib. 
V 1.402 - 1.0 ) 

The net work is 15,600 - 2107 - 2430 = 11,063 ft.-lb. The heat expenditure in 
this case is Jl(T E - T c ) = 17.72 x (1537 - 153) = 24,500 ft.-lb., and the efficiency 
is 11,063 -=- 24,500 = 0.453; considerably greater than that of either the Lenoir or 
the Brayton engine (14). If we express the cyclic area as 100, then that of the 
Lenoir engine is 52 and that of the Brayton engine is 104. 

304. Trial Results. These comparisons correspond with the consumption of 
gas found in actual practice with the three types of engine. The three efficiencies 
are 0.226, 0.366, "and 0.453. Taking 4 cu. ft. of free gas as ideally capable of giv- 
ing one horse power per hour, the gas consumption per hp.-hr. in the three cases 
would be respectively 4 - 0.226 = 17.7, 4 - 0.366 = 10.9, and 4 -- 0.453 = 8.84 cu. ft. 
Actual tests gave for the Lenoir and Hugon engines 90 cu. ft. ; for the Brayton, 
50 ; and for the modified Otto, 21. The possibility of a great increase in economy 



164 



APPLIED THERMODYNAMICS 



by the use of an engine of a form somewhat similar to that of the Brayton will be 
discussed later. 



305. Complete Pressure Cycle. The cycle of Art. 303 merits detailed exami- 
nation. In Fig. 129, the heat absorbed is l(T E — T c ) ; that rejected is 

KT F -T G )+k(T G -T B )', 
the efficiency is 

To - T» 



1 - 



TV- T ( 



T E -Tc y T E - T c 

The entropy diagram may be drawn as ebmnd, Fig. 124, showing this cycle to be 
more efficient than the equal-length-stroke Otto cycle, but less efficient than the 
Atkinson. With complete expansion down to the lower pressure limit, the cycle 
becomes BCEFH, Fig. 129, or ebod, Fig. 124; the strokes are still of unequal 
length, and the efficiency is (Fig. 129) 

J T E - T c 

If the strokes be made of equal length, with incomplete expansion, T G =T B , the 
cycle becomes the ordinary Otto, and the efficiency is 

TV - T n T c - T n 



1- 



r i\ 



T 



306. Oil Engines : The Diesel Cycle. Oil engines may operate in either 
the two-stroke or the four-stroke cycle, usually the latter ; and combus- 
tion may occur at constant volume (Otto), constant pressure (Brayton), or 
constant temperature (Diesel). Diesel, in 1893 (15), first proposed what 
has proved to be from a thermal standpoint the most economical heat 
engine. It is a four-cycle engine, approaching more closely than the 
Otto to the Carnot cycle, and theoretically applicable to solid, liquid, or 

gaseous fuels, although actually used only 
with oil. The first engine, tested by Schroter 
in 1897, gave indicated thermal efficiencies 
ranging from 0.34 to 0.39 (16). The ideal- 
ized cycle is shown in Fig. 130. The opera- 
tions are adiabatic compression, isothermal 
expansion, adiabatic expansion, and dis- 




Fig. 130. 



Arts. 306, 307. — Diesel 
Cycle. 



charge at constant volume. Pure air is com- 
pressed to a high pressure and temperature, 
and a spray of oil is then gradually injected by means of external air 
pressure. The temperature of the cylinder is so high as at once to ignite 
the oil, the supply of which is so adjusted as to produce combustion 
practically at constant temperature. Adiabatic expansion occurs after 
the supply of fuel is discontinued. A considerable excess of air is used. 
The pressure along the combustion line is from 30 to 40 atmospheres, that 



THE DIESEL ENGINE 



165 



at which the oil is delivered is 50 atmospheres, and the temperature at 
the end of compression approaches 1000° F. The engine is started by 
compressed air ; two or more cylinders are used. There is no uncertainty 
as to the time of ignition ; it begins immediately upon the entrance of 




FlG. 131. Art. 306. — Diesel Engine. (American Diesel Engine Company.) 



the oil into the cylinder. To avoid pre-ignition in the supply tank, the 
high-pressure air used to inject the oil must be cooled. The cylinder 
is water-jacketed. Figure 131 shows a three-cylinder engine of this type ; 
Fig. 132, its actual indicator diagram, reversed. 



166 



APPLIED THERMODYNAMICS 




Fig. 132. Art. 306. — Indicator Diagram, Diesel Engine. 
(16 x 24 in. engine, 160 r.p.ui. Spring 400.) 

307. Efficiency. The heat absorbed along ab, Fig. 130, is 

'a * a 

The heat rejected along/t? is l(T s — T d ~). We may write the efficiency 



as 



1 __ l(T f - T t ) . 

* a 



^v) ; wlience 



But T f = Tjffi" 1 - r o (^y _1 ,and T a =T c 

For the heat rejected along fd we may therefore write 



y 



;©"-} 



and for the efficiency, 



hT A 



1- 



..Va 






yRT a log, 



v. 



v b 



This increases as T a increases and as —r decreases. The last conclu- 

* a 

sion is of prime importance, indicating that the efficiency should in- 
crease at light loads. This may be apprehended from the entropy 
diagram, abfd, Fig. 124. As the width of the cycle decreases (bf 
moving toward acT), the efficiency increases. 



MODIFICATIONS OF THE OTTO CYCLE 



167 



In constructing the entropy diagram from an actual Diesel indicator card 
a difficulty arises similar to one met with in steam engine cards; the quantity of 
substance in the cylinder is not constant (Art. 454). This has been discussed 
by Eddy (17), Frith (18), and Reeve (19). 
The illustrative diagram, constructed as 
in Art. 347, is suggestive. Figure 133 
shows such a diagram, for an engine 
tested by Denton (20). The initially hot 
cylinder causes a rapid absorption of heat 
from the walls during the early part of 
compression along ab. Later, along he, 
heat is transferred in the opposite direc- 
tion. Combustion occurs along cd, the 
temperature and quantity of heat increas- 
ing rapidly. During expansion, along de, 
the temperature falls with increasing 
rapidity, the path becoming practically adiabatic during release, along ef. The 
TV diagram of Fig. 133 indicates that no further rise of temperature would ac- 
company increased compression ; the actual path at y has already become prac- 
tically isothermal. 




Fig. 133. 



"= VOR N 

Art. 307.— Diesel Engine 
Diagrams. 



308. Comparison of Cycles. Figure 134 shows all of the cycles that have 
been discussed, on a single pair of diagrams. The lettering corresponds 
with that in Figs. 122-128, 130. The cycles are, 

Carnot, abed, Lenoir, df g h ,dfQi , Diesel, dabf, 

Otto, ebfd, Bray ton, dnbgh, dnbi, Atkinson, ebed, 

Complete pressure, debgh, debi. 





Fig. 134. Art. 308, Probs. 7, 25. — Comparison of Gas Engine Cycles. 



Practical Modifications of the Otto Cycle 

309. Importance of Proper Mixture. The working substance used in gas 
engines is a mixture of gas, oil vapor or oil, and air. Such mixtures will not 
ignite if too weak or too strong. Even when so proportioned as to permit of 
ignition, any variation from the ideally correct ratio has a detrimental effect: if 



168 APPLIED THERMODYNAMICS 

too little air is present, the gas will not burn completely, the exhaust will be dark- 
colored and odorous, and unburned gas may explode in the exhaust pipe when 

it meets more air. If too much air is admitted, 
the products of combustion will be unnecessarily 
diluted and the rise of temperature during 
ignition will be decreased, causing a loss of work 
area on the PV diagram. Figure 135 shows the 
effect on rise of temperature and pressure of 
varying the proportions of air and gas, assuming 
the variations to remain within the limits of 




Fig. 135. Art. 309. - Effect of Possible ignition. Failure to ignite may occur 
Mixture Strength as a res ult of the presence of excess of air as 

well as when the air supply is deficient. Rapidity 
of flame propagation is essential for efficiency, and this is only possible with a 
proper mixture. The gas may in some cases burn so slowly as to leave the cyl- 
inder partially unconsumed. In an engine of the type shown in Fig. 119, this 
may result in a spread of flame through /, B, and C back to D, with dangerous 
consequences. 

310. Methods of Mixing. The constituents of the mixture must be intimately 
mingled in a finely divided state, and the governing of the engine should prefer- 
ably be accomplished by a method which keeps the proportions at those of highest 
efficiency. Variations of pressure in gas supply mains may interpose serious dif- 
ficulty in this respect. Fluctuations in the lights which may be supplied from the 
same mains are also excessive as the engine load changes. Both difficulties are 
sometimes obviated in small units by the use of a rubber supply receiver. Varia- 
tions in the speed of the engine often change the proportions of the mixture. 
When the air is drawn from out of doors, as with automobile engines, variations 
in the temperature of the air affect the mixture composition. In simple types of 
engine, the relative openings of the automatic gas and air inlet valves are fixed 
when the engine is installed, and are not changed unless the quality or pressure 
of the gas changes, when a new adjustment is made by the aid of the indicator or 
by observation of the exhaust. Mechanically operated valves are used on high- 
speed engines ; these are positive in their action. The use of separate pumps for 
supplying air and gas permits of proportioning in the ratio of the pump displace- 
ments, the volume delivered being constant, regardless of the pressure or tempera- 
ture. Many adjustable mixing valves and carburetors are made, in which the 
mixture strength may be regulated at will. These are necessary where irregulari- 
ties of pressure or temperature occur, but require close attention for economical 
results. The presence of burned gas in the clearance space of the cylinder affects 
the mixture, retarding the flame propagation. The effect of mixture strength on 
allowable compression pressures remains to be considered. 

311. Actual Gas Engine Diagram. A typical indicator diagram from 
a good Otto cycle engine is shown in Fig. 136. The various lines differ 
somewhat from those established in Art. 288. These differences we now 
discuss. Figure 137 shows the portion bcde of the diagram in Fig. 136 
to an enlarged vertical scale, thus representing the action more clearly. 



ALLOWABLE COMPRESSION 169 

The line fg is that of atmospheric pressure, omitted in Fig. 136. We will 
begin our study of the actual cycle with the compression line. 





Fig. 136. Arts. 311, 342, 345.— Fig. 137. Arts. 311, 326, 328. — En- 

Otto Engine Indicator Diagram. larged Portion of Indicator Diagram. 

312. Limitations of Compression. It has been shown that a high degree 
of compression is theoretically essential to economy. In practice, com- 
pression must be limited to pressures (and corresponding temperatures) 
at which the gases will not ignite of themselves; else combustion will 
occur before the piston reaches the end of the stroke, and a backward 
impulse will be given. Gases differ widely as to the temperatures at 
which they will ignite; hydrogen, for example, inflames so readily that 
Lucke (21) estimates that the allowable final pressure must be reduced 
one atmosphere for each 5 per cent of hydrogen present in a mixed gas. 

The following are the average final gauge compression pressures 

recommended by Lucke (22) : for gasoline, in automobile engines, 

45 to 95 lb., in ordinary engines, 60 to 85 lb. ; for kerosene, 30 to 85 lb.; 

for natural gas, 75 to 130 lb. ; for coal gas or carbureted water gas, 

60 to 100 lb. ; for 'producer gas y 100 to 160 lb. ; and for blast furnace 

gas, 120 to 190 lb. The range of compression depends also upon the 

pressure existing in the cylinder at the beginning of compression ; for 

two-cycle engines, this varies from 18 to 21 lb., and for four-cycle 

engines, from 12 to 14 lb., both absolute. 

The pre-compression temperature also limits the allowable range below the 
point of self-ignition. This temperature is not that of the entering gases, but it 
is that of the cylinder contents at the moment when compression begins ; it is 
determined by the amount of heat given to the incoming gases by the hot cylin- 
der walls, and this depends largely upon the thoroughness of the water jacketing 
and the speed of the engine. This accounts for the rather wide ranges of allow- 
able compression pressures above given. Usual pre-compression temperatures are 
from 140° to 300° F. li Scavenging" the cylinder with cold air, the injection of 
water, or the circulation of water in tubes in the clearance space, may reduce this. 
Usual practice is to thoroughly jacket all exposed surfaces, including pistons 
and valve faces, and to avoid pockets where exhaust gases may collect. The 
primary object of jacketing, however, is to keep the cylinder cool, both for me- 
chanical reasons and to avoid uncontrollable explosions at the moment when the 
gas reaches the cylinder. 



170 APPLIED THERMODYNAMICS 

313. Practical Advantages of Compression. Compression pressures have 
steadily increased since 1881, and engine efficiencies have increased correspond- 
ingly, although the latter gain has been in part due to other causes. Improved 
methods of ignition have permitted of this increased compression. Besides the 
thermodynamic advantage already discussed, compression increases the engine 
capacity. In a non-compressive engine, no considerable range of expansion could 
be secured without allowing the final pressure to fall too low to give a large work 
area; in the compressive engine, wide expansion limits may be obtained along 
with a fairly high terminal pressure. Compression reduces the exposed cylinder 
surface in proportion to the weight of gas present at maximum temperature, and 
so decreases the loss of heat to the walls. The decreased proportion of clearance 
space following the use of compression also reduces the proportion of spent gases 
to be mixed with the incoming charge. 

314. Pressure Rise during Combustion. In Art. 292, the pressure P h after 
combustion was assumed. While, for reasons which will appear, any computation 
of the rise of pressure by ordinary methods is unreliable, the method should be 
described. Let H denote the amount of heat liberated by combustion, per pound 

of fuel. Then, Fig. 122, H= l(T h - T e ), T b -T e =- and T h = — + T e . But 
P± = Ih = E- + 1. Then P h - P e = *¥*. But — e = —, whence 

p e T e ny ir e T e v; 

P e _ k-l _ k-l 
lT e TvT~ l~' 

'(g) 5 * 

H(k - /) 0.402 H 



Then P b - P e 



PJ d 



V. 



315. Computed Maximum Temperature. Dealing now with the constant 
volume ignition line of the ideal diagram, let the gas be one pound of pure 
carbon monoxide, mixed with just the amount of air necessary for com- 
bustion (2.48 lb.), the temperature at the end of compression being 1000° 
absolute, and the pressure 200 lb. absolute. Since the heating value of 1 
lb. of CO is 4315 B. t. u., while the specific heat at constant volume of 
C0 2 is 0.1692, that of N being 0.1727, we have 

rise in temperature * (1 . 57 x q^+^I X 0.1727) = 72C6 ° F " 
The temperature after complete ignition is then 8265° absolute. The 
pressure is 200 x ^^ = 1653 lb. If the volume increases during igni- 
tion, the pressure decreases. Suppose the volume to be doubled, the rise 
of temperature being, nevertheless, as computed : then the maximum pres- 
sure attained is 826.5 lb. 



DISSOCIATION 171 

316. Actual Maxima. No such, temperature as 8265° absolute is at- 
tained. In actual practice, the temperature after ignition is usually about 
3500° absolute, and the pressure under 400 lb. The rise of either is less 
than half of the rise theoretically computed, for the actual air supply, 
with the actual gas delivered. It is difficult to measure the maximum tem- 
perature, on account of its extremely brief duration. It is more usual to 
measure the pressure and compute the temperature. This is best done by 
a graphical method, as with the indicator. 

317. Explanation of Discrepancy. There are several reasons for the disagree- 
ment between computed and observed results. Charles' law does not hold rigidly 
at high temperatures; the specific heats of gases are known to increase with the 
temperature (Meyer found in one case the theoretical maximum temperature to 
be reduced from 4250° F. to 3330° F. by taking account of the increases in specific 
heats as determined by Mallard and Le Ch atelier); combustion is actually not 
instantaneous throughout the mass of gas and some increase of volume always 
occurs; and the temperature is lowered by the cooling effect of the cylinder walls. 
Still another reason for the discrepancy is suggested in Art. 318. 

318. Dissociation. Just as a certain maximum temperature must be attained 
to permit of combustion, so a certain maximum temperature must not be exceeded 
if combustion is to continue. If this latter temperature is exceeded, a suppression 
of combustion ensues. Mallard and Le Chatelier found this " dissociation " effect 
to begin at about 3200° F. with carbon monoxide and at about 4500° F. with steam. 
Deville, however, found dissociative effects with steam at 1800° F., and with car- 
bon dioxide at still lower temperatures. The effect of dissociation is to produce, 
at each temperature within the critical range for the gas in question, a stable 
ratio of combined to elementary gases, — e.g. of steam to oxygen and hydrogen, — 
which cannot widely vary. No exact relation between specific temperatures and 
such stable ratio has yet been determined. It has been found, however, that the 
maximum temperature actually attained by the combustion of hydrogen in oxygen 
is from 3500° to 3800° C, although the theoretical temperature is about 9000° C. 
At constant pressure (the preceding figures refer to combustion at constant vol- 
ume), the actual and theoretical figures are 2500° and 6000° C. respectively. For 
hydrogen burning in air, the figures are 1830 to 2000°, and 3800° C. Dissociation 
here steps in to limit the complete utilization of the heat in the fuel. In gas en- 
gine practice, the temperatures are so low that dissociation cannot account for all 
of the discrepancy between observed and computed values ; but it probably plays 
a part. 

319. Rate of Flame Propagation. This has been mentioned as a factor influ- 
encing the maximum temperature and pressure attained. The speed at which 
flame travels in an inflammable mixture, if at rest, seldom exceeds 65 ft. per sec- 
ond. If under pressure or agitation, pulsations may be produced, giving rise to 
" explosion waves," in which the velocity is increased and excessive variations in 
pressure occur, as combustion is more or less localized (23). Clerk (24), experi- 



172 



APPLIED THERMODYNAMICS 



meriting on mixtures of coal gas with air, found maximum pressure to be obtained 
in minimum time when the proportion of air to gas by volume was 5 or 6 to 1 '. 
for pure hydrogen and air, the best mixture was 5 to 2. The Massachusetts Insti- 
tute of Technology experiments, made with carbureted water gas, showed the best 
mixture to be 5 to 1; with 86° gasoline, the quickest inflammation was obtained 
when 0.0217 parts of gasoline were mixed with 1 part of air; with 76° gasoline, 
when 0.0263 to 0.0278 parts were used.* Grover found the best mixture for coal 
gas to be 7 to 1 ; for acetylene, 7 or 8 to 1, acetylene giving higher pressures than 
coal gas. With coal gas, the weakest ignitible mixture was 15 to 1, the theoreti- 
cally perfect mixture being 5.7 to 1. The limit of weakness with acetylene was 18 
to 1. Both Grover and Lucke (26) have investigated the effect of the presence of 
u neutrals " (carbon dioxide and nitrogen, derived either from the air, the incom- 
ing gases, or from residual burnt gas) on the rapidity of propagation. The re- 



o 45 















07 5 














70.1 




68.7 


NO NEUTRAL ^^ 




70.4 














ONE 
74.0 


PART NEUTRAL Ann 


ED 


74^-^ 






72.1 
PARTS NEUTRAL ADt 


74.0 






76.0 




77.6 ^pE 


|75.7 | . 

c PARTS NEUTRAL ADDED 


1t2 

79J3 




77.3 


ITparts NE 


UTKAL ADD 


ED 


79.6 






S0.6 








80.5 




81.3 




81 = 3 



3.5 



4.5 5 5.5 

PARTS AIR PER ONE PART GAS 



6.5 



Fig. 138. Art. 319. —Effect of Presence of Neutrals. 
(From Hutton's " The Gas Engine," by permission of John Wiley & Sons, Publishers.) 

suits of Lucke's study of water gas are shown in Fig. 138. The ordinates show 
the maximum pressures obtained with various proportions of air and gas. These 
are highest, for all percentages of neutral, at a ratio of air to gas of 5 to 1 ; but 
they decrease as the proportion of neutral increases. The experiments indicate 
that the speed of flame travel varies widely with the nature of the mixture and the 
conditions of pressure to which it is subjected. If the mixture is too weak or too 
strong, it will not inflame at all. 

320. Piston Speed. The actual shape of the ideally vertical ignition line will 
depend largely upon the speed of flame propagation as compared with the speed 
of the piston. Figure 139, after Lucke, illustrates this. The three diagrams were 
taken from the same engine under exactly the same conditions, excepting that the 
speeds in the three cases were 150, 500, and 750 r. p. m. Similar effects may be 
obtained by varying the mixture (and consequently the flame speed) while keep- 
ing the piston speed constant. High compression causes quick ignition. Throt- 



* The theoretical ratio of air to C 6 Hi 4 is 47 to 1. 



IGNITION 



173 



tling of the incoming charge increases the percentage of neutral from the burnt 
gases and retards ignition. 

























160 
80 










M.E. 


P. 48.80 


















^ 










































1 













150 r. p. m. 









M.E.P. 39 


.84 












120 
80 
40 




































, 










J 

























500 r. p. m. 





























M 


.E.P. 


28.96 
















80 
40 


\ 






















> 























750 r. p. m. 

Fig. 139. Art. 320. — Ignition Line as affected by Piston Speed. 

(From Lucke's " Gas Engine Design.") 



321. Point of Ignition. The spreading of flame is at first slow. Ignition is, 
therefore, made to occur prior to the end of the stroke, giving a practically verti- 
cal line at the end, where inflammation is well under way. Figure 140, from 
Poole (27), shows the effects of change in the point of ignition. In (a) and (/>), 
ignition was so early as to produce a negative loop on the diagram. This was cor- 
rected in (c), but (d) represents a still better diagram. 'In (e) and (/), ignition 
was so late that the comparatively high piston speed kept the pressure down, and 
the work area was small. It is evident that too early a point of ignition causes a 
backward impulse on the piston, tending to stop the engine. Even though the 
inertia of the fly wheel carries the piston past its " dead point," a large amount of 
power is wasted. The same loss of power follows accidental pre-ignition, whether 
due to excessive compression, contact with hot burnt gases, leakage past piston 
rings, or other causes. Failure to ignite causes loss of capacity and irregularity 



174 



APPLIED THERMODYNAMICS 



of speed, but theoretically at least does not affect economy. For reasons already 
suggested, light loads (where governing is effected by throttling the supply) and 
weak mixtures call for early ignition. 




IGNITION 20% EARLY 




IGNITION 16% EARLY 







Fig. 140. Art. 321. — Time of Ignition. 
(From Poole's "The Gas Engine," by permission of the Hill Publishing Company.) 



322. Methods of Ignition. An early method for igniting the gas was to use 
an external flame enclosed in a rotating chamber which at proper intervals opened 
communication between the flame and the gas. This arrangement was applicable 
to slow speeds only, and some gas always escaped. In early Otto engines, the 
external flame with a sliding valve was used at speeds as high as 100 r. p. m. (28). 
The insertion periodically of a heated plate, once practiced, was too uncertain. 
The use of an internal flame, as in the Brayton engine, was limited in its applica- 
tion and introduced an element of danger. Self-ignition by the catalytic action 
of compressed gas upon spongy platinum was not sufficiently positive and reliable. 
The use of an incandescent wire, electrically heated and mechanically brought 
into contact with the gas, was a forerunner of modern electrical methods. The 
"hot tube" method is still in frequent use, particularly in England. This in- 
volves the use of an externally heated refractory tube, which is exposed to the gas 
either intermittently by means of a timing valve, or continuously, ignition being 
then controlled by adjusting the position of the external flame. In the Hornsby- 
Akroyd and Diesel engines, ignition is self-induced by compression alone ; but 
external heating is necessary to start these engines. 



IGNITION AND EXPANSION 



175 



323. Electrical Methods. The two modern electrical methods are the 
" make -and break " and " jump spark." In the former, an electric current, 
generated from batteries or a small dynamo, is passed through two sepa- 
rable contacts located in the cylinder and connected in series with a spark 
coil. At the proper instant, the contacts are separated and a spark passes 
between them. In the jump spark system, an induction coil is used and 
the contacts are stationary. A series' of sparks is thrown between them when 
the primary circuit is closed, just before the end of the compression stroke. 

324. Clearance Space. The combustion chamber formed in the clearance 
space must be of proper size to produce the desired final pressure. A common 
ratio to piston displacement is 30 per cent. Hutton has shown (29) that the 
limits for best results may range easily from 8.7 to 56 per cent (Art. 332) . 

325. Expansion Curve. Slow inflammation has been shown to result in a de- 
creased maximum pressure after ignition. Inflammation occurring during expan- 
sion as a result of slow spreading of the flame is called " after burning." Ideally, 
the expansion curve should be adiabatic ; actually it falls in most cases above the 
air adiabatic, pv 1A0 ' 2 = constant, although it is known that during expansion from 40 
to 50 per cent of the total heat in the gas is being 
carried away by the jacket icater. Figure 141 repre- 
sents an extreme case; after burning has made the 
expansion line almost horizontal, and some unburnt 
gas is being discharged to the exhaust. Those who 
hold to the dissociation theory would explain this 
line on the ground that the gases dissociated during combustion are gradually 
combining as the temperature falls ; but actually, the temperature is not falling, 
and the effect which we call after burning is most pronounced with weak mix- 
tures and at such low temperatures as do not permit of any considerable 
amount of dissociation. Practically, dissociation has the same effect as an 
increasing specific heat at high temperature. It affects the ignition line to 

some extent ; but the shape of the expansion line is to a far greater de- 
gree determined by the slow inflammation of the gases. The effect of 
the transfer of heat between the fluid and the cylinder walls is dis- 
cussed in Art. 347. The actual exponent of the expansion 
curve varies from 1.2 in large engines to 1.38 in good small 
engines, occasionally, however, rising as high as 1.5. 
The compression curve usually, though 
not always, has a slightly 
higher exponent. The 
adiabatic exponent for a 
mixture of hydrocarbon 
gases is lower than that 
for air or a perfect gas ; and in some cases the actual adiabatic, plotted for the 
gases used, would be above the determined expansion line, as should normally be 
expected, in spite of after burning. The presence of explosion waves (Art. 319) 
may modify the shape of the expansion curve, as in Fig. 142. The equivalent 




Fig 



Art. 325. — After 
Burning:. 




Fig. 142. Art. 325. — Explosion Waves. 



176 



APPLIED THERMODYNAMICS 




Fig. 143. Art. 



326. — Delayed Exhaust Valve 
Opening. 



curve may be plotted as a mean through the oscillations. Care must be taken 
not to confuse these vibrations with those due to the inertia of the indicating 

instrument. 

326- The Exhaust Line. This is 
shown to an enlarged vertical scale 
as be, Fig. 137. " Low spring " dia- 
grams of this form are extremely use- 
ful. As engines wear, more or less 
" lost motion " becomes present in the 
valve-actuating gear, and the tendency of this is to vary the instant of opening 
or closing the inlet or the exhaust valve. The effect of delayed opening of the 
latter is shown in Fig. 143 ; that 
of an inadequate exhaust passage, 
in Fig. 144. An early opening 
of the exhaust valve may cause 
loss also, as in Fig. 145. In mul- 
tiple cylinder engines having com- 
mon exhaust and suction mains, 
early exhaust from one cylinder 




Fig. 144. Art. 326. 



Throttled Exhaust Passages. 

may produce a rise of pressure during the latter part of the exhaust stroke of 
another. Obstructions to suction and discharge movements of gas are com- 
monly classed together as 
" fluid friction." This may in 
small engines amount to as 
much as 30 per cent of the 
power developed. In good 
engines of large or moderate 
size, it should not exceed 6 per 
cent. It increases, propor- 
tionately, at light loads ; and 




Fig. 145. Art. 326. — Exhaust Valve Opening too Early. 

possibly absolutely as well if governing is effected by throttling the charge. 



327. Scavenging. To avoid the presence of burnt gases in the clear- 
ance space, and their subsequent mingling with the fresh charge, " scav- 
enging," or sweeping out these gases from the cylinder, is sometimes prac- 
ticed. This may be accomplished by means of a separate air pump, or by 
adding two idle strokes to the four strokes of the Otto cycle. In the 
Crossley engines, the air admission valve was opened before the gas valve, 
and before the termination of the exhaust stroke. By using a long ex- 
haust pipe, the gases were discharged in a rather violent puff, which pro- 
duced a partial vacuum in the cylinder. This in turn caused a rush of 
air into the clearance space, which swept out the burnt gases by the time 
the piston had reached the end of its stroke. Scavenging decreases the 
danger of missing ignitions with weak gas, tends to prevent pre-ignition, 
and appears to have reduced the consumption of fuel. 



DIAGRAM FACTOR 



177 



328. The Suction Stroke. This also is shown in Fig. 137, line cd. The effect 
of late opening of the valve is shown in Fig. 146 ; that of an obstructed passage 
or of throttling the supply, in Fig. 
147. If the opening is too early, 
exhaust gases will enter the supply 
pipe. If closure is too early, the 
gas will expand during the re- 
mainder of the suction stroke, but 
the net work lost is negligible; if 
too late, some gas will be discharged 
back to the supply pipe during the 
beginning of the compression stroke, 
as in Fig. 148. Excessive obstruc- 
tion in the suction passages de- 
creases the capacity of the engine, 
in a way already suggested in the 
study of air compressors (Art. 224). 




Art. 328. — Delayed Opening of 
Suction Valve. 




ACTUAL 

Fig. 147. Art. 328. — Throttled Suction. 




Fig. 148. Art, 328. — Late Closing of 
Suction Valve. 



329. Diagram Factor. The 

discussion of Art. 309 to Art. 
328 serves to show why the 
work area of any actual dia- 
gram must always be less than 
that of the ideal diagram for 
the same cylinder, as given in 
Fig. 122. The ratio of the 
two is called the diagram 

factor. The area of the ideal card would constantly increase as 
compression increased ; that of the actual card soon reaches a limit 
in this respect; and, consequently, in general, the diagram factor 
decreases as compression increases. Variations in excellence of 
design are also responsible for variations of diagram factor. 

In the best recorded tests, its value has ranged from 0.38 to 0.59 ; in 
ordinary practice, the values given by Lucke (30) are as follows: for 
kerosene, if previously vaporized and compressed, 0.30 to 0.40, if injected 
on a hot tube, 0.20; for gasoline, 0.52 to 0.50; for producer gas, 0. 40 to 
0.56 ; for coal gas, 0.45 ; for carbureted water gas, 0.45 ; for blast furnace 
gas, 0.30 to 0.48 ; for natural gas, O.4O to 0.52. These figures are for four- 
cycle engines. For two-cycle engines, usual values are about 20 per cent 
less. Figure 149 shows on the PV and entropy planes an actual indicator 
diagram with the corresponding ideal cycle. 



178 



APPLIED THERMODYNAMICS 





ACTUAL DIAGRAM 
IDEAL DIAGRAM 



Fig. 149. Art. 329. —Actual and Ideal Gas Engine Diagrams. 

Gas Engine Design 

330. Capacity. The work done per stroke may readily be computed for the 
ideal cycle, as in Art. 293. This may be multiplied by the diagram factor to 
determine the probable performance of an actual engine. To develop a given 
power, the number of cycles per minute must be established. Ordinary piston 
speeds are from 450 to 1000 ft. per minute, usually lying between 550 and 800 ft., 
the larger engines having the higher speeds. The stroke ranges from 1.0 to 2.0 
times the diameter, the ratio increasing, generally, with the size of the engine. 
A gas engine has no overload capacity, strictly speaking, since all of the factors 
entering into the determination of its capacity are intimately related to its effi- 
ciency. It can be given a margin of capacity by making it larger than the 
computations indicate as necessary, but this or any other method involves a con- 
siderable sacrifice of the economv at normal load. 



331. Mean Effective Pressure. Since in an engine of given size the extreme 
volume range of the cycle is fixed, the mean net ordinate of the ivork area measures 
the capacity. The quotient of the cycle area by the volume range gives what is called 
the mean effective pressure (m. e. p.). In Fig. 122, it is ebfd +(V d — V e ). We 

then write m. e. p. = W -f- (J d - V e ); but from Art. 295, W = Q[l - {^j * ] ; Q 

being the gross quantity of heat absorbed in the cycle. Then, in proper units, 
without allowance for diagram factor, 



m. e. p. = 



«Mg)'] 



332. Illustrative Problem. To determine the cylinder dimensions of a four-cycle, 
two-cylinder, double-acting engine of 500 hp., using producer gas {assumed to contain 



GAS ENGINE DESIGN 



179 



CO, 394; N, 60; H, 0.6; parts in 100 by weight) (Art. 285), at 150 r. p. m. and a 
piston speed of 825 ft. per minute. 

We assume (Fig. 150), P 1 = 12, P 2 = 114.7, T x = 200° F., and diagram factor 
= 0.48 (Arts 312, 329). 



Since P x l\» = P 2 V 2 y, 



Zi-[p*Y 



To 



ffi'-m- 



5.9. Let the piston displace- 



ment V 1 — V-2 



I). Then F 2 = 0.2045 D and F, = 1.2045 D. The clearance is 



0.2045 (Art. 324)^ 



Also T-2 



TiPzVt 659.6 x 144.7 x 0.2045 



135- 



B v ' PiVi 12 x 1.2045 

absolute. The heat evolved per pound of the mixed gas (taking the calorific 

value of hydrogen burned to steam as 53,400) is (0.394 x 4315) + (0.006 x 53,400) 

= 2021 B. t. u. The products of com- p 
bustion consist of ff x 0.394 = 
0.619 lb. of C0 2 (specific heat = 0.1692), 
0.006 x 9 = 0.054 lb. of H 2 (steam, 
specific heat 0.37), and || (0.619 — 
0.394) = 0.751 lb. of N" accompanying 
the oxygen introduced to burn the 
CO, with (0.054-0.006)|| = 0.1607 lb. 
of X accompanying the oxygen in- 
troduced to burn the H ; and 0.60 lb. 
of N originally in the gas, making a 
total of 1.5117 lb. of N (specific heat 
0.1727). To raise the temperature of 
these constituents 1° F. at constant 




Fig. 150. 



Arts. 332-335. 
Engine. 



Design of Gas 



volume requires (0.619 x 0.1692) + (0.054x0.37) + (1.5117 x 0.1727) = 0.3849 
B. t. u. The rise in temperature T 3 - T 2 is then 2021 - 0.3849 = 5260°, and 
T z = 5260 + 1357 = 6617° absolute. Then 



P 2 ^ = 144.7 



6617 



and 



The work per cycle is 



P,V, 



Pi 



T 2 

P, 



= 709> 



12 



144.7 



1357 

° = 58.7. 



^4-^2+^1 



r- 141x0 18 pf X 709 x 0.2045) - (58.7 x 1.2045) -(144.7 x 0.2045) + (12 x 1.2045) 1 
L 0.402 J 

= 10,080 D foot pounds. 

In a two-cylinder, four-cycle, double-acting engine, all of the strokes are work- 
ing strokes ; the foot-pounds of work per stroke necessary to develop 500 hp. are 



* While the use of a " blanket " diagram factor as in this illustration may be justi- 
fied, in any actual design the clearance at least must be ascertained from the actual 
exponent of the compression curve. The design as a whole, moreover, would better 
be based on special assumptions as in Problem 15, (b), page 197. 



180 APPLIED THERMODYNAMICS 

— - — — — — = 55,000. The necessary piston displacement per stroke, D, is 
2, X loO 

55,000 - 10,080 = 5.46 cu. ft. The stroke is 825 -=- (2 x 150) = 2.75 ft. or 33 in. The 
piston area is then 5.46 -=-2.75 = 1.985 sq. ft. or 285.5 sq. in. The area of the water- 
cooled tail rod may be about 33 sq. in., so that the cylinder area should be 
285.5 + 33 = 318.5 sq. in. and its diameter consequently 20.14 in. 

333. Modified Design. In an actual design for the assumed conditions, over- 
load capacity was secured by assuming a load of 600 hp. to be carried with 20 per 
cent excess air in the mixture. (At theoretical air supply, the power developed 
should then somewhat exceed 600 hp.) The air supply per pound of gas is now 

[(0.394 x if) + (0.006x8)] ^xl.l= 1.422 lb. 

Of this amount, 0.23 x 1.422 = 0.327 lb. is oxygen. The products of combustion 
are f| x 0.394 = 0.619 lb. C0 2 , 0.006 x 9 = 0.054 lb. H 2 0, (1.422 - 0.327) + 0.60 
= 1.693 lb. N, and 0.327- (if x 0.394) - (8 x 0.006) = 0.054 lb. of excess oxygen ; a 
total of 2.422 lb. The rise in temperature T 3 - T 2 is 

?0?1 = 47G0O< 

(0.619 x 0.1692) + (0.054 x 0.37) + (1.693 x 0.1727) + (0.054 x 0.1551) 
Then T s = 4760 + 1357 = 61 17° absolute, 

• ^=^^ = 144.71111 = 655,^ = ^^ = 12^=54.2, 

and the work per cycle is 

i/m <\aq nf~ (655 x 0.2045) - (54.2 x 1.2045) - (144.7 x 0.2045) + (12 x 1.2045) 1 
144 x U.4»i^ ; 5^02 J 

= 9150 D foot-pounds. 

The piston displacement per stroke is = 7.21 cu. ft., the cylinder 

* * * 2 x 150 x 9150 9 

area is (7.21-^-2.75)144 + 33 = 410 sq. in., and its diameter 22.83 in. The cylinders 

were actually made 23| by 33 in., the gas composition being independently assumed. 

334. Estimate of Efficiency. To determine the 'probable efficiency of the engine 
under consideration : each pound of working substance is supplied with 1.422 lb. 
of air. Multiplying the weights of the constituents by their respective specific 
volumes, we obtain as the volume of mixture per pound of gas, 31.33 cu. ft. at 
14.7 lb. pressure and 32° F., as follows : — 

CO, 0.394 x 12.75 = 5.01 

H, 0.006 x 178.83 = 1.07 

N. 0.600 x 12.75 = 7.65 

Air, 1.422 x 12.387 = 17.60 

31.33 

At the state 1, Fig. 150, 7\ = 659.6, P x = 12, whence 

y = T^P a F n _ 659.6 x 14.7 x 31.33 = gl 2 
1 P X T 12 x 491.6 

The piston displaces 7.21 x 300 = 2163 cu. ft. or 2163 - 51.2 = 42.3 lb. of this mix- 
ture per minute. The heat taken in per minute is then 2021 x 4:2.3 = 85,200 B. t. u. 



AUTOMOBILE ENGINE RATING 181 

The work done per minute is 600 x 880Q0 = 25500 B. t. u. The efficiency is then 
25,500 -4- 85,200 = 0.299. An actual test of the engine gave 0.282, with a load 
somewhat under 600 hp. The Otto cycle efficiency is 1857 ~ 6 ° 9,6 = 0.516. 

335. Automobile Engine. To ascertain the probable capacity and economy of a 

four-cylinder, four-cycle, single-acting gasoline engine with cylinders 4- by 5 in., at 

8 

1500 r. p. m. 

In Fig. 150, assume P, = 12, P 2 = 84.7, T x = 70° F., diagram factor, 0.375 
(Arts. 312, 329). Assume the heating value of gasoline at 19,000 B. t. u., and its 
composition as C 6 H U : its vapor density as 3.05 (air = 1.). Let the theoretically 
necessary quantity of air be supplied. 

The engine will give two cycles per revolution. Its active piston displacement 

7854 x (4-) 2 x 5 
is then — v sJ x 3000 = 145.5 cu. ft. per minute, which may be repre- 
sented as V 1 — V 2 , Fig. 150. We now find 

Zs = (i2_V' 713 = 0.2495; F 2 = 0.2495 TV, 0.7505 7i = 145.5; Fi = 194; F 2 = 48.5; 
V 1 V84.7/ 

Clearance = 48 ' 5 = 0.334 (Art. 324) ; T 2 = M ' 7 X 48 ' 5 X 529 ' 6 = 936° absolute. 
145.5 12 x 194 

To burn one pound of gasoline there are required 3.53 lb. of oxygen, or 15.3 lb. 
of air. For one cubic foot of gasoline, we must supply 3.05 x 15.3 = 46.6 cu. ft. 
of air. The 145.5 cu. ft. of mixture displaced per minute must then consist of 
145.5 -r- 47.6 = 3.06 cu. ft. of gasoline and 142.44 cu. ft. of air, at 70° F. and 12 lb. 

pressure. The specific volume of air at this state is 5 ' 29 ' 6 x 14 -7 x 12.387 = lg 3g 
F F 491.6 x 12 

cu. ft. ; that of gasolene is 16.38 -=- 3.05 = 5.37 cu. ft. The iceight of gasoline 
used per minute is then 3.06 -=- 5.37 = 0.571 lb. The heat used per minute is 
0.571 x 19,000 = 10,840 B. t. u. The combustion reaction may be written 

C 6 H 14 + 19 = 6C0 2 -f 7H 2 
86 + 304 = 264 + 126 

V£ = 3.O6 lb. C0 2 per lb. C 6 H 14 
L2_6 = 1.35 ib. H 2 per lb. C 6 H ]4 
II x -W- = H.82 lb. N per lb. C 6 H 14 

16.23 = 1. + 15.3, approximately. 

The heat required to raise the temperature of the products of combustion 1° F. is 
[(3.06x0.1692) + (1.35 x 0.37) + (11.82 x 0.1727)] 0.571 = 1.646 B. t. u. per minute. 
The rise in temperature T* - T 2 is then 10,840 - 1.646 = 6610°, T 3 = 6610 + 936 

=7546° absolute, P 3 = 84.7 1^ = 681, P 4 = 12 ■?**!= 96.7, and the work per minute is 

936 84.7 r 

375 x 144 [" (681 x 48.5)- (96.7 x 194) - (84.7 x 48.5) + (12 x 194) 1 = 1 

L 0.402 J ' ' 

foot-pounds. This is equivalent to ^|2ffl« = 2160 B. t. u. per minute or to 
^"llo^ff- = 51 horsepower. The efficiency is 2160 -h 10840 = 0.^0. In an automobile 



182 



APPLIED THERMODYNAMICS 



running at 50 miles per hour, this would correspond to 50-4- (0.571 x 60) = 1.46 miles 
run pei' pound of gasoline. In practice, the air supply is usually deficient, and the 
power and economy less than those computed. 

It is obvious that with a given fuel, the diagram factor and other data of 
assumption are virtually fixed. An approximation of the power of the engine 
may then be made, based on the piston displacement only. This justifies in some 
measure the various rules proposed for rating automobile engines (30 a). 

Current Gas Engine Forms 

336. Otto Cycle Oil Engines. This class includes, among many others, the 
Mietz and Weiss, two-cycle, and the Daimler, Priestman, and Hornsby-Akroyd, 
four-cycle. In the last named, shown in Fig. 151, kerosene oil is injected by a 




Fig. 151. Art. 336. — Kerosene Engine with Vaporizer. 
(From " The Gas Engine," by Cecil P. Poole, with the permission of the Hill Publishing Company.) 



small pump into the vaporizer. Air is drawn into the cylinder during the suction 
stroke, and compressed into the vaporizer on the compression stroke, where the 
simultaneous presence of a critical mixture and a high temperature produces the 
explosion. External heat must be applied for starting. The point of ignition is 
determined by the amount of compression ; and this may be varied by adjusting 
the length of the connecting rod on the valve gear. The engine is governed by 
partially throttling the charge of oil, thus weakening the mixture and the force 
of the explosion. The oil consumption may be reduced to less than 1 lb. per 
brake hp. per hour. 



TYPES OF GAS ENGINE 



183 



In the Priestman engine, an earlier type, air under pressure sprayed the oil 
into a vaporizer kept hot by the exhaust gases. The method of governing was to 
reduce the quantity of charge without changing its proportions. A hand pump 
and external heat for the vaporizer were necessary in starting. An indicated 
thermal efficiency of 0.165 has been obtained. The Daimler (German) engine 
uses hot-tube ignition without a timing valve, the hot tube serving as a vaporizer. 
Extraordinarily high speeds are attained. 

337. Modern Gas Engines : the Otto. The present-day small Otto engine is ordi- 
narily single-cylinder and single-acting, governing on the " hit or miss " principle 
(Art. 343). It is used with all kinds of gas and with gasoline. Ignition is elec- 
trical, the cylinder water jacketed, the jackets cast separately from the cylinder. 
The Foos engine, a simple, compact form, often made portable, is similar in princi- 
ple. In the Crossley-Otto, a leading British type, hot-tube ignition is used, and 
the large units have two horizontal opposed single-acting cylinders. In the 
Andrews form, tandem cylinders are used, the two pistons being connected by 
external side rods. 

338. The Westinghouse Engine. This has recently been developed in very 
large units. Figure 152 shows the working side of a two-cylinder, tandem, 
double acting engine, representing the inlet valves on top of the cylinders. 




Fig. 152. Arts. 338, 350. —Westinghouse Gas Engine. Two-cylinder Tandem, Four-cycle. 



Smaller engines are often built vertical, with one, two, or three single-acting 
cylinders. All of these engines are four-cycle, with electric ignition, governing 
by varying the quantity and proportions of the admitted mixture. Sections of 
the cylinder of the Riverside horizontal, tandem, double-acting engine are shown in 
Fig. 153. It has an extremely massive frame. The Allis-Chalmers engine is built 
in large units along similar general lines. Thirty-six of the latter engines of 
4000 hp. capacity each on blast furnace gas are now (1909) being constructed. 
They weigh, each, about 1,500,000 lb., and run at 83 1 r. p. m. The cylinders are 
44 by 54 in. Nearly all are to be direct-connected to electric generators. 



184 



APPLIED THERMODYNAMICS 




to 
3 






:=*■■'. 



TYPES OF GAS ENGINE 185 

339. Two-cycle Engines. In these, the explosions are twice as frequent as 
with the four-cycle engine, and cooling is consequently more difficult. With an 
equal number of cylinders, single- or double-acting, the two-cycle engine of course 
gives better regulation. The first important two-cycle engine was introduced by 
Clerk in 1880. The principle was the same as that of the engine shown in Fig. 119. 
The Oechelhaueser engine has two single-acting pistons in one cylinder, which are 
connected with cranks at 180°, so that they alternately approach toward and 
recede from each other. The engine frame is excessively long. Changes in the 
quantity of fuel supplied control the speed. The Koerting engine, a double-acting 
horizontal form, has two pumps, one for air and one for gas. A "scavenging" 
charge of air is admitted just prior to the entrance of the gas, sweeping out the 
burnt gases and acting as a cushion between the incoming charge and the exhaust 
ports. The engine is built in large units, with electrical ignition and compressed 
air starting gear. The speed is controlled by changing the mixture proportions. 

340. Special Engines. For motor bicycles, a single air-cooled cylinder is often 
used, with gasoline fuel. Occasionally, two cylinders are employed. The engine 
is four-cycle and runs at high speed. Starting is effected by foot power, which 
can be employed whenever desired. Ignition is electrical and adjustable. The 
speed is controlled by throttling. Extended surface air-cooled cylinders have also 
been used on automobiles, a fan being employed to circulate the air, but the limit 
of size appears to be about 7 hp. per cylinder. Most automobiles have water- 
cooled cylinders, usually four in number, four-cycle, single-acting, running at 
about 1000 to 1200 r. p. m., normally. Governing is by throttling and by chang- 
ing the point of ignition. The cylinders are usually vertical, the jacket water 
being circulated by a centrifugal pump, and being used repeatedly. Both hot-tube 
and electrical methods of ignition have been employed, but the former is now 
almost wholly obsolete. The number of cylinders varies from one to six ; occa- 
sionally they are arranged horizontally, duplex, or opposed. Two-cycle engines 
have been introduced. The fuel in this country is usually gasoline. For launch 
engines, the two-cycle principle is popular, the crank case forming the pump 
chamber, and governing being accomplished by throttling. Kerosene or gasoline 
are the fuels. 

341. Alcohol Engines. These are used on automobiles in France. A special 
carburetor is employed. The cylinder and piston arrangement is sometimes that 
of the Oechelhaueser engine (Art. 339). The speed is controlled by varying the 
point of ignition. In launch applications, the alcohol is condensed, on account of 
its high cost, and in some cases is not burned, but serves merely as a working fluid 
in a " steam " cylinder, being alternately vaporized by an externally applied gaso- 
line flame and condensed in a surface condenser. The low value of the latent 
heat of vaporization (Art. 360) of alcohol permits of "getting up steam" more 
rapidly than is possible in an ordinary steam engine. 

342. Basis of Efficiency. The performances of gas engines may be compared 
by the cubic feet of gas, or pounds of liquid fuel, or pounds of coal gasified in the 
producer, per horse power hour ; but since none of these figures affords any really 



186 APPLIED THERMODYNAMICS 

definite basis, oil account of variations in heating value, it is usual to express the 
results of trials in heat units consumed per horse power per minute. Since one horse 
power equals 33,000 +- 778 = 42.42 B. t. u. per minute, this constant divided by the 
heat unit consumption gives the indicated thermal efficiency. In making tests, the 
over-all efficiency of a producer plant may be ascertained by weighing the coal. 
When liquid fuel is used, the engine efficiency can readily be determined separately. 
To do this with gas involves the measurement of the gas, always a matter of some 
difficulty with any but small engines. The measurement of power by the indicator 
is also inaccurate, possibly to as great an extent as 5 per cent, which may be reduced to 
2 per cent, according to Hopkinson, by employing mirror indicators. This error has 
resulted in the custom of expressing performance in heat units consumed per brake 
horse power per hour or per kw.-hr., where the engines are directly connected to 
generators. There is some question as to the proper method of considering the 
negative loop, bcde, of Fig. 136. By some, its area is deducted from the gross work 
area, and the difference used in computing the indicated horse power. By others, 
the gross work area of Fig. 136 is alone considered, and the " fluid friction " losses 
producing the negative loop are then classed with engine friction as reducing the 
"mechanical efficiency." Various codes for testing gas engines are in use (31). 

343. Typical Figures. Small oil or gasoline engines may easily show 10 per 
cent brake efficiency. Alcohol engines of small size consume less than 2 pt. per 
brake hp.-hr. at full load (32). A well-adjusted Otto engine has given an indicated 
thermal efficiency of 0.19 with gasoline and 0.23 with kerosene (33). Ordinary 
power gas engines of average size under test conditions have repeatedly shown 
indicated thermal efficiencies of 25 to 29 per cent. A Cockerill engine gave 30 per 
cent. Hubert (34) tested at Seraing an engine showing nearly 32 per cent indicated 
thermal efficiency. Mathot (35) reports a test of an Ehrhardt and Lehmer double- 
acting, four-cycle 600 hp. engine at Heinitz which reached nearly 38 per cent. A 
blast furnace gas engine gave at full load 25.4 per cent. Expressed in pounds of 
coal, one plant with a low load factor gave a kilowatt-hour per 1.8 lb. In another 
case, 1.59 lb. was reached, and in another, 2.97 lb. of wood per kw.-hr. It is common 
to hear of guarantees of 1 lb. of coal per brake hp.-hr., or of 11,000 B. t. u. in gas. 
A recent test of a Crossley engine is reported to have shown the result 1.13 lb. of 
coal per kw.-hr. Under ordinary running conditions, 1.5 to 2.0 lb. with varying 
load may easily be realized. These latter figures are of course for coal burned in 
the producer. They represent the joint efficiency of the engine and the producer. 
The best results have been obtained in Germany. For the engine alone, Schroter 
is reported to have obtained on a Guldner engine an indicated thermal efficiency of 
0.427 at full load with illuminating gas (36). 

The efficiency cannot exceed that of the ideal Otto cycle. In one test of an 
Otto cycle engine an indicated thermal efficiency of 0.37 was obtained, while the 
ideal Otto efficiency was only 0.41. The engine was thus within 10 per cent of 
perfection for its cycle. 

The Diesel engine has given from 0.32 to 0.412 indicated thermal efficiency. 
Its cycle, as has been shown, permits of higher efficiency than that of Otto. 

344. Plant Efficiency. Figures have been given on coal consumption. Over- 
all efficiencies from fuel to indicated work have ranged from 0.14 upward. At the 



GAS ENGINE TRIALS 187 

Maschinenfabrik Winterthur, a consumption of 0.7 lb. of coal (13,850 B. t. u.) per 
brake hp.-hr. at full load has been reported (37). This is closely paralleled by the 
0.285 plant efficiency obtained on the Guldner engine mentioned in Art. 343 when 
operated with a suction producer on anthracite coal. At the Royal Foundry, 
Wurtemburg (38), 0.78 lb. of anthracite were burned per Ihp.-hr., and at the 
Imperial Post Office, Hamburg, 0.93 lb. of coke. In the best engines, variations of 
efficiency with reasonable changes of load below the normal have been greatly 
reduced, largely by improved methods of governing. 

345. Mechanical Efficiency. The ratio of work at the brake to net indicated 
work ranges about the same for gas as for steam engines having the same arrange- 
ment of cylinders. When mechanical efficiency is understood in this sense, its 
value is nearly constant for a given engine at all loads, decreasing to a slight 
extent only as the load is reduced. In the other sense, suggested in Art. 342, i.e. 
the mechanical efficiency being the ratio of work at the brake to gross indicated 
work (no deduction being made for the negative loop area of Fig. 136), its value 
falls off sharply as the load decreases, on account of the increased proportion of 
"fluid friction." Lucke (39) gives the following as average values for the 
mechanical efficiency in the latter sense : — 



Engine 


Mechanical Efficiency 




Four-cycle 


Two-cycle 


Large, 500 Ihp. and over, 

Medium, 25 to 500 Ihp., . . . . 
Small, 4 to 25 Ihp., 


0.81 to 0.86 
0.79 to 0.81 
0.74 to 0.80 


0.63 to 0.70 
0.64 to 0.66 
0.63 to 0.70 



346. Heat Balance. The principal losses in the gas engine are due to 
the cooling action of the jacket water (a necessary evil in present practice) 
and to the heat carried away in the exhaust. The arithmetical means of 
nine trials collated by the writer give the following percentages represent- 
ing the disposition of the total heat supplied: to the jacket, 40.52; to 
the exhaust, 33.15 ; work, 21.87 ; unaccounted for, 6.23. Hutton (40) 
tabulates a large number of trials, from which similar arithmetical aver- 
ages are derived as follows: to the jacket, 37.96; to the exhaust, 29.84; 
work; 22.24 ; unaccounted for, 8.6. In general, the larger engines show a 
greater proportion of heat converted to work, an increased loss to the 
exhaust, and a decreased loss to the jacket. 

347. Entropy Diagram. When the PV diagram is given, points may be trans- 

v p 

ferred to the entropy plane by the formula n b — n a = &log e —^+ l\og e -^ (Art. 

169). The state a may be taken at 32° F. and atmospheric pressure ; then the 
entropy at any other state b depends simply upon V b and P b . To find F a > we 
must know the equation of the gas. According to Richmond (41), the mean 



188 



APPLIED THERMODYNAMICS 



value of k may be taken at 0.246 on the compression curve and at 0.26 on the ex- 
pansion curve, while the mean values of I corresponding are 0.176 and 0.189. The 
values of R are then 778(0.216 - 0.176)= 54.46 and 778(0.260 - 0.189) = 55.24. 
The characteristic equations are, then, PV = 54.46 T along the compression curve; 
and PV — 55.24 T along the expansion curve. The formula gives changes of en- 
tropy per pound of substance. The indicator diagram does not ordinarily depict 
the behavior of one pound; but if the weight of substance used per cycle be 
known, the volumes taken from the PV diagram may be converted to specific 
volumes for substitution in the formula. 

It is sometimes desirable to study the TV relations throughout the cycle. In 
Fig. 154, let ABCD be the PV diagram. Let EF be any line of constant volume 
intersecting this diagram at G, H. By Charles' law, T G : T H ::P G : P H . The 



PqrT 




Fig. 154. Art. 347. — Gas Engine TV Diagram. 



ordinates JG, JH may therefore serve to represent temperatures as well as pres- 
sures, to some scale as yet undetermined. If the ordinate JG represent tempera- 
ture, then the line OG is a line of constant pressure. Let the pressure along this 
line on a TV diagram be the same as that along IG on a, PV diagram. Then 
(again by Charles' law) the line OH is a line of constant pressure on the jTFplane, 
corresponding to the line KH on the PV plane. Similarly, OL corresponds to 
MN and OQ to RB. Project the points S, T, R, B, where MN and RB intersect 
the PV diagram, until they intersect OL, OQ. Then points U, Q, W, X are 
points on the corresponding TV diagram. The scale of T is determined from 
the characteristic equation; the value of R may be taken at a mean between 
the two given. A transfer may now be made to the NT plane by the aid of the 

equation n b - n a = I log e -£ + (k - l)\og e ^ (Art. 169), in which T a = 491.6, 

_ 54.46 x 491.6 _° 
Va " 2116.8 ~ 1 - b4 ' 



GOVERNING 



189 



Figure 155, from Reeve (42), is from a similar four-cycle engine. The enor- 
mous area BA CD represents heat lost to the water jacket. The inner dead center 
of the engine is at x ; thereafter, for a short A 

period, heat is evidently abstracted from the 
fluid, being afterward restored, just as in the 
case of a steam engine (Art. 431), because 
during expansion the temperature of the gases 
falls below that of the cylinder w T alls. Reeve 
gives several instances in which the expansive 
path resembles xBzD ; other investigators find 
a constant loss of heat during expansion. Fig- 
ure 156 gives the PV and NT diagrams for 
a Hornsby-Akroyd engine; the expansion 
line be here actually rises above the isothermal, 
indicative of excessive after burning. 

348. Methods of Governing. The power 
exerted by an Otto cycle engine may 
be varied in accordance with the external 
load, by various methods ; in order that 
efficiency may be maintained, the govern- 
ing should not lower the ratio of pressures during compression. To ensure 
this, variation of the clearance, by mechanical means or water pockets and 
outside compression have been proposed, but no practicably efficient means 

T 




Fig. 155. Art. 347. — Gas Engine 
Entropy Diagram. 





■N 



Fig. 15G. Art. 347. — Diagrams for Hornsby-Akroyd Engine. 



have yet been developed. Automobile engines are often governed by 
varying the point of ignition, a most wasteful method, because the reduc- 
tion in power thus effected is unaccompanied by any change whatever in 
fuel consumption. Equally wasteful is the use of excessively small ports 
for inlet or exhaust, causing an increased negative loop area and a conse- 
quent reduction in power when the speed tends to increase. In engines 



190 



APPLIED THERMODYNAMICS 




where the combustion is gradual, as in the Brayton or Diesel, the point of 
cut-off of the charge may be changed, giving the same sort of control as in 
a steam engine. 

Three methods of governing Otto cycle engines are in more or less 
common use. In the "hit-or-miss" plan, the engine omits drawing in its 
charge as the external load decreases. One or more idle strokes ensue. 
No loss of economy results (at least from a theoretical standpoint), but the 
speed of the engine is apt to vary on account of the increased irregularity 
of the already occasional impulses. Governing by changing the proportions 
of the mixture (the total amount being kept constant) should apparently 
not affect the compression; actually, however, the compression must be 
p fixed at a sufficiently low point to 

avoid danger of pre-ignition to the 
strongest probable mixture, and 
thus at other proportions the de- 
gree of compression will be less 
than that of highest efficiency. A 
change in the quantity of the mix- 
ture, without change in its propor- 
tions, by throttling the suction or 
by entirely closing the inlet valve 
toward the end of the suction 
stroke, results in a decided change 
of compression pressure, the superimposed cards being similar to those 
shown in Fig. 157. In theory, at least, the range of compression pressures 
would not be affected; but the variation in proportion of clearance gas 
present requires injurious limitations of final compression pressure, just 
as when governing is effected by variations in mixture strength. 

349. Defects in Gas Engine Governing. The hit-or-miss system may be 
regarded as entirely inapplicable to large engines. The other practicable 
methods sacrifice the efficiency. Further than this, the governing influ- 
ence is exerted during the suction stroke, one full revolution (in four- 
cycle engines) previous to the working stroke, which should be made equal 
in effort to the external load. If the load changes during the intervening 
revolution, the control will be inadequate. Gas engines tend therefore to 
irregularity in speed and low efficiency under variable or light loads. The 
first disadvantage is overcome by increasing the number of cylinders, the 
weight of the fly wheel, etc., all of which entails additional cost. The sec- 
ond disadvantage has not yet been overcome. 



Fig. 157. Art. 348. — Effect of Throttling. 



350. Construction Details. The irregular impulses characteristic of the gas 
engine and the high initial pressures attained require excessively heavy and 



DETAILS 191 

strong frames. For anything like good regulation, the fly wheels must also be 
exceptionally heavy. For small engines, the bed casting is usually a single heavy 
piece. The type of frame usually employed on large engines is illustrated in Fig. 
152. It is in contact with the foundation for its entire length, and in many cases 
is tied together by rods at the top extending from cylinder to cylinder. 

Each working end of the cylinder of a four-cycle engine must have two valves, 
— one for admission and one for exhaust. In many cases, three valves are used, 
the air and gas being admitted separately. The valves are of the plain disk or 
mushroom type, with beveled seats ; in large engines, they are sometimes of the 
double-beat type, shown in Fig. 153. Sliding valves cannot be employed at the 
high temperature of the gas cylinder. Exhaust opening must always be under 
positive control ; the inlet valves may be automatic if the speed is low, but are 
generally mechanically operated on large engines. In horizontal four-cycle 
engines, a cam shaft is driven from an eccentric at half the speed of the engine. 
Cams on this shaft operate each of the controlling valves by means of adjustable 
oscillating levers, a supplementary spring being employed to accelerate the closing 
of the valves. In order that air or gas may pass at constant speed through the 
ports, the cam curve must be carefully proportioned w r ith reference to the varia- 
tion in conditions in the cylinder (13). Hutton (44) advises proportioning of 
ports such that the mean velocity may not exceed 60 ft. per second for automatic 
inlet valves, 90 ft. for mechanically operated valves, and 75 ft. for exhaust valves, 
on small engines. 

351. Starting Gear. No gas engine is self-starting. Small engines are often 
started by turning the fly wheel by hand, or by the aid of a bar or gearing. An 
auxiliary hand air pump may also be employed to begin the movement. A small 
electric motor is sometimes used to drive a gear -faced fly wheel with which the 
motor pinion meshes. In all cases, the engine starts against its friction load only, 
and it is usual to provide a method of keeping the exhaust valve open during part 
of the compression stroke so as to decrease the resistance. In multiple-cylinder 
engines, as in automobiles, the ignition is checked just prior to stopping. A com- 
pressed but unexploded charge will then often be available for restarting. In the 
Clerk engine, a supply of unexploded mixture was taken during compression from 
the cylinder to a strong storage tank, from which it could be subsequently drawn. 
Gasoline railway motor cars are often started by means of a smokeless powder 
cartridge exploded in the cylinder. Modern large engines are started by com- 
pressed air, furnished by a direct-driven or independent pump, and stored in small 
tanks. 

352. Jackets. The use of water-spray injection during expansion has been 
abandoned, and air cooling is practicable only in small sizes. The cylinder, 
piston, piston rod, and valves must usually be thoroughly water-jacketed. Posi- 
tive circulation must be provided, and the water cannot be used over again unless 
artificially cooled. At a heat consumption of 200 B. t. u. per minute per I hp., 
with a 40 per cent loss to the jacket, the theoretical consumption of water heated 
from 80 to 160° F. is exactly 1 lb. per Ihp. per minute. This is greater than the 
water consumption of a non-condensing steam plant, but much less than that- of 



192 APPLIED THERMODYNAMICS 

a condensing plant. The discharge water from large engines is usually kept 
below 130° F. In smaller units, it may leave the jackets at as high a temperature 
as 160° F. 

353. Possibilities of Gas Power. The gas engine, at a comparatively early 
stage in its development, has surpassed the best steam engines in thermal effi- 
ciency. Mechanically, it is less perfect than the latter; and commercially it is 
regarded as handicapped by the greater reliability, more general field of applica- 
tion, and much lower cost (excepting, possibly, in the largest sizes *) of the steam 
engine. The use of producer gas for power eliminates the coal smoke nuisance ; 
the stand-by losses of producers are low; and gas may be stored, in small quanti- 
ties at least. The small gas engine is quite economical and may be kept so. The 
small steam engine is usually wasteful. The Otto cycle engine regulates badly, a 
disadvantage which can be overcome at excessive cost; it is not self -starting ; the 
cylinder must be cooled. Even if the mechanical necessity for jacketing could be 
overcome, the same loss would be experienced, the heat being then carried off in 
the exhaust. The ratio of expansion is too low, causing excessive waste of heat 
at the exhaust, which, however, it may prove possible to reclaim. The heat in the 
jacket water is large in quantity and low in temperature, so that the prob- 
lem of utilization is confronted with the second law of thermodynamics. 
Methods of reversing have not yet been worked out, and no important marine 
applications of gas power have been made, although small producer plants have 
been installed for ferryboat service with clutch reversal, and compressed and 
stored gas has been used for driving river steamers in France, England, and 
Germany. 

The proposed combinations of steam and gas plants, the gas plant to take the 
uniform load and the steam units to care for fluctuations, really beg the whole 
question of comparative desirability. The bad " characteristic " curve — low effi- 
ciency at light loads and absence of bona fide overload capacity — will always bar 
the gas engine from some services, even where the storage battery is used as an 
auxiliary. Many manufacturing plants must have steam in any case for process 
work. In such, it will be difficult for the gas engine to gain a foothold. For the 
utilization of blast furnace waste, even aside from any question of commercial 
power distribution, the gas engine has become of prime economic importance. 

(1) Hutton, The Gas Engine, 1908, 545 ; Clerk, Theory of the Gas Engine, 1903, 
75. (2) Hutton, The Gas Engine, 1908, 158. (3) Clerk, The Gas Engine, 1890, 
119-121. (4) Ibid., 129. (5) Ibid., 133. (6) Ibid.. 137. (7) Ibid., 198. (8) En- 
gineering News, October 14, 1906, 357. (9) Lucke and Woodward, Tests of Alcohol 
Fuel, 1907. (10) Junge, Poiver, December, 1907. (10 a) For a fuller exposition of the 
limits of producer efficiency with either steam or waste gas as a diluent, see the author's 
paper, Trans. Am. Inst. Chem. Engrs., Vol. IT. (11) Trans. A S. M. E., XXVIII, 6, 
1052. (12) A test efficiency of 0.657 was obtained by Parker, Holmes, and Campbell : 
United States Geological Survey, Professional Paper No. 48. (13) Ewing, The Steam 
Engine, 1906, 418. (14) Clerk, The Theory of the Gas Engine, 1903. (15) Theorie 
und Construction eines rationellen Warmemotors. (16) Zeuner, Technical Thermody- 
namics (Klein), 1907, I, 439. (17) Trans. A. S. 31. E., XXI, 275. (18) Ibid, 286. 

* Piston speeds of large gas engines may exceed those of steam engines. 



GAS POWER 193 

(19) Op. cit., XXIV, 171. (20) Op. cit., XXI, 276. (21) Gas Engine Design, 1897, 
33. (22) Op. cit., p. 34 et seq. (23) See Lucke, Trans. A. 8. M. E., XXX, 4, 418. 
(24) The Gas Engine, 1890, p. 95 et seq. (25) A. L. Westcott, Some Gas Engine Cal- 
culations based on Fuel and Exhaust Gases : Power, April 13, 1909, p. 693. (26) Hut- 
ton, The Gas Engine, 1908, pp. 507, 522. (27) The Gas Engine, 1908. (28) Clerk, 
op. cit., p. 216. (29) Op. cit., p. 291. (30) Op. cit., p. 38. The corresponding usual 
mean effective pressures are given on p. 36. (30a) See the author's paper, Commer- 
cial Eatings for Internal Combustion Engines, in Machinery, April, 1910. (31) Zeits. 
Ver. Deutsch. Ing., November 24, 1906 ; Power, February, 1907. (32) The Electrical 
World, December 7, 1907, p. 1132. (33) Trans. A. S. M. E., XXIV, 1065. (34) Bui. 
Soc. de V Industrie Mineral, Ser. Ill, XIV, 1461. (35) Trans. A. S. M. E., XXVIII, 
6, 1041. (36) Quoted by Mathot, supra. (37) Also from Mathot. (38) Mathot, 
supra. (39) Op. cit., p. 5. (40) Op. cit., pp. 342-343. (41) Trans. A. S. M. E., XIX, 
491. (42) Ibid., XXIV, 171. (43) Lucke, Gas Engine Design, 1905. (44) Op. cit., 
483. 

SYNOPSIS OF CHAPTER XI 
The Producer 

The importance of the gas engine is largely due to the producer process for making 

cheap gas. 
In the gas engine, combustion occurs in the cylinder, and the highest temperature 

attained by the substance determines the cyclic efficiency. 
Fuels are natural gas, carbureted and uncarbureted water gas, coal gas, coke oven 

gas, producer gas, blast furnace gas ; gasoline, kerosene, fuel oil, distillate, 
alcohol, coal tars. 
The gas producer is a lined cylindrical shell in which the fixed carbon is converted 

into carbon monoxide, while the hydrocarbons are distilled off, the necessary heat 

being supplied by the fixed carbon burning to CO. 
The maximum theoretical efficiency of the producer making power gas is less than that 

of the steam boiler. Either steam or exhaust gas from the engine must be intro- 
duced to attain maximum efficiency. 
The mean composition of producer gas, by volume, is CO, 19.2; C0 2 , 9.5 ; H, 12.4 ; 

CH 4 , C 2 H 4 , 3.1 ; N, 55.8. 
The " figure of merit " is the heating value of the gas fter pound of carbon contained. 



Gas Engine Cycles 

The Otto cycle is bounded by two adiabatics and two lines of constant volume ; the 

engine may operate in either the four-stroke cycle or the two-stroke cycle. 
In the two-stroke cycle, the inlet and exhaust ports are both open at once. 

In the Otto cycle, — 6 = -^ and — 5 = ^. 
P e P d T e T d 

Efficiency = e ~ Td = 1 - ( — j"7" = h ~ f = 1 — t—A ~v~\ it depends solely on the 

extent of compression. 

Efficiency of Atkinson engine {isothermal rejection of heat) = 1 — — — log e — * ; 

higher than that of the Otto cycle. 6 — « e 



194 APPLIED THERMODYNAMICS 

Lenoir cycle : constant pressure rejection of heat; efficiency =1— _J? zA^y^A ±* 

T f - T d T f — T d ' 

T g — Ti, Th— Td . 



y(T b -T n ) T b -T n 



Brayton cycle : combustion at constant pressure ; efficiency = 1 — 

•ji 'j' 

or, with complete expansion, — _ *• 

A special comparison shows the Clerk Otto engine to give a much higher efficiency than 
the Brayton or Lenoir engine, but that the Brayton engine gives slightly the largest 
work area. 

The Clerk Otto (complete pressure) cycle gives an efficiency of 1 — f ~ g . v s ~ h 

T e - Tc y T e -T c ' 

intermediate between that of the ordinary Otto and the Atkinson. 



as ratio of expansion decreases. yBT a log e — 



The Diesel cycle: isothermal combustion; efficiency =1 — — ; increases 



Modifications in Practice 

The PV diagram of an actual Otto cycle engine is influenced by 

(a) proportions of the mixture, which must not be too weak or too strong, and 
must be controllable ; 

(&) maximum allowable temperature after compression to avoid pre-ignition ; the 
range of compression, which determines the efficiency, depends upon this as 
well as upon the pre-compression pressure and temperature ; 

(c) the rise of pressure and temperature during combustion; always less than 

those theoretically computed, on account of (1) divergences from Charles' 
law, (2) the variable specific heats of gases, (3) slow combustion, (4) disso- 
ciation ; 

(d) the shape of the expansion curve, usually above the adiabatic, on account of 

after burning, in spite of loss of heat to the cylinder wall; 

(e) the forms of the suction and exhaust lines, which may be affected by badly 

proportioned ports and passages and by improper valve action. 
Dissociation prevents the combustion reaction of more than a certain proportion of 

the elementary gases at each temperature within the critical limits. 
The point of ignition must somewhat precede the end of the stroke, particularly with 

weak mixtures. 
Methods of ignition are by hot tube, jump spark, and make and break. 
Cylinder clearance ranges from 8.7 to 56 per cent. It is determined by the compression 

pressure range. 
Scavenging is the expulsion of the burnt gases in the clearance space prior to the 

suction stroke. 
The diagram factor is the ratio of the area of the indicator diagram to that of the ideal 

cycle. 



Mean effective pressure — 



V e 



GAS POWER 195 

Gas Engine Design 

In designing an engine for a given power, the gas composition, rotative 
speed and piston speed are assumed. The probable efficiency may be 
estimated in advance. Overload capacity must be secured by assum- 
ing a higher capacity than that normally needed ; the engine will do 
no more work than that for which it is designed. 

Current Forms 

Otto cycle oil engines include the Mietz and Weiss, two-cycle, and the Daimler, Priest- 
man, and Hornsby-Akroyd, four-cycle. 

Modern forms of the Otto gas engine include the Otto, Foos, Crossley-Otto, and 
Andrews. 

The Westinghouse, Riverside, and Allis-Chalmers engines are built in the largest sizes. 

Two-cycle gas engines include the Oechelhaueser and Koerting. 

Special engines are built for motor bicycles, automobiles, and launches, and for burn- 
ing alcohol. 

The basis of efficiency is the heat unit consumption per horse power per minute. 

The mechanical efficiency may be computed from either gross or net indicated work. 

Becorded efficiencies of gas engines range up to ^.7 per cent ; plant efficiencies to 0.7 
lb. coal per brake hp.-hr. 

The mechanical efficiency increases with the size of the engine, and is greater with the 
four-stroke cycle. 

About 38 per cent of the heat supplied is carried off by the jacket water, and about 
33 per cent by the exhaust gases, in ordinary practice. 

The entropy diagram may be constructed by transfer from the PVor TV diagrams. 

Governing is effected 

(a) by the hit-or-miss method; economical, but unsatisfactory for speed regulation, 



(&) by throttling, 

(c) by changing mixture proportions, 



both wasteful. 



In all cases, the governing effort is exerted too early in the cycle. - 
Gas engines must have heavy frames and fly wheels; exhaust valves (and inlet valves 
at high speed) must be mechanically operated by carefully designed cams ; pro- 
vision must be made for starting ; cylinders and other exposed parts are jacketed. 
About 1 lb. of jacket water is required per Ihp. -minute. 
Gas engine advantages : high thermal efficiency ; elimination of coal smoke nuisance ; 
stand-by losses are low ; gas may be stored ; economical in small units ; desirable 
for utilizing blast furnace gas. 
Disadvantages : mechanically still evolving ; of unproven reliability ; less general field 
of application ; generally higher first cost ; poor regulation ; not self-starting ; 
cylinder must be cooled ; low ratio of expansion ; non-reversible ; no overload 
capacity ; no available by-product heat for process work in manufacturing plants. 

PROBLEMS 

1. Compute the volume of air ideally necessary for the complete combustion of 
1 cu. ft. of gasoline vapor, C 6 H 14# 



196 APPLIED THERMODYNAMICS 

2. Find the maximum theoretical efficiency, using pure air only, of a power gas 
producer fed with a fuel consisting of 70 per cent of fixed carbon and 30 per cent of 
volatile hydrocarbons. 

3. In Problem 2, what is the theoretical efficiency if 20 per cent of the oxygen 
necessary for gasifying the fixed carbon is furnished by steam ? 

4. In Problem 3, if the hydrocarbons (assumed to pass off unchanged) are half 
pure hydrogen and half marsh gas, compute the producer gas composition by volume, 
using specific volumes as follows : nitrogen, 12.75 ; hydrogen, 178.83 ; carbon mo- 
noxide, 12.75 ; marsh gas, 22.3. 

5. A producer gasifying pure carbon is supplied with the theoretically necessary 
amount of oxygen from the atmosphere and from the gas engine exhaust. The latter 
consists of 28.4 per cent of C0 2 and 71.6 per cent of N, by weight, and is admitted to 
the extent of 1 lb. per pound of pure carbon gasified. Find the rise in temperature, 
the composition of the produced gas, and the efficiency of the process. The heat of 
decomposition of C0 2 to C may be taken at 14,500 B. t. u. per pound of carbon. 

6. Find the figures of merit in Problems 4 and 5. (Take the heating value of H 
at 53,400 ; of CH 4 , at 22,500.) 

7. In Fig. 134, let ^ = 4, P d = 30, P 9 = P gn = P d + 10, T h = 3000°, T d = 1000° 

(absolute). Find the efficiency and area of each of the ten cycles, for 1 lb. of air, with- 
out using efficiency formulas. 

8. In Problem 7, show graphically by the NT diagram that the Carnot cycle is 
the most efficient. 

9. What is the maximum theoretical efficiency of an Otto four-cycle engine in 
which the fuel used is producer gas ? (See Art. 312.) 

10. What maximum temperature should theoretically be attained in an Otto en- 
gine using gasoline, with a temperature after compression of 780° F. ? (The heat liber- 
ated by the gasoline, available for increasing the temperature, may be taken at 19,000 
B. t. u. per pound.) 

11. Find the mean effective pressure and the work done in an Otto cycle between 
volume limits of 0.5 and 2.0 cu. ft. and pressure limits of 14.7 and 200 lb. per square 
inch absolute. 

12. An Otto engine is supplied with pure CO, with pure air in just the theoretical 
amount for perfect combustion. Assume that the dissociation effect is indicated by the 
formula* (1.00 — a) (6000 — T) = 300, in which a is the proportion of gas that will 
combine at the temperature T° F. If the temperature after compression is 800° F., 
what is the maximum temperature attained during combustion, and what proportion 
of the gas will burn during expansion and exhaust, if the combustion line is one of con- 
stant volume ? The value of I for CO is 0.1758. 

13. An Otto engine has a stroke of 24 in., a connecting rod 60 in. long, and a pis- 
ton speed of 400 ft. per minute. The clearance is 20 per cent of the piston displace- 
ment, and the volume of the gas, on account of the speed of the piston as compared 
with that of the flame, is doubled during ignition. Plot its path on the PV diagram 

* This is assumed merely for illustrative purposes. It has no foundation and is irra- 
tional at limiting values. 



GAS POWER 197 

and plot the modified path when the piston speed is increased to 800 ft. per minute, 
assuming the flame to travel at uniform speed and the pressure to increase directly as 
the spread of the flame. The pressure range during ignition is from 100 to 200 lb. 

14. The engine in Problem 11 is four-cycle, two-cylinder, double-acting, and makes 
100 r. p. m. with a diagram factor of 0.40. Find its capacity. 

15. Starting at T d = 14.7, V d = 43.45, T d = 32° F. (Fig. 122), plot (a) the ideal 
Otto cycle for 1 lb. of CO with the necessary air, and (5) the probable actual cycle 
modified as described in Arts. 309-328, and find the diagram factor. Clearance is 25 
per cent of the piston displacement in both cases. 

16. Find the cylinder dimensions in Art. 332 if the gas composition be as given in 
Art. 285. (Take the average heating value of CII 4 and C 2 H 4 at 22,500 B. t. u. per pound, 
and assume that the gas contains the same amount of each of these constituents.) 

17. Find the clearance, cylinder dimensions, and probable efficiency in Art. 332 if 
the engine is two-cycle. 

18. Find the size of cylinders of a four-cylinder, four-cycle, single-acting gasoline 
engine to develop 30 bhp. at 1200 r. p. m., the cylinder diameter being equal to the 
stroke. Estimate its thermal efficiency, the theoretically necessary quantity of air 
being supplied. 

19. An automobile consumes 1 gal. of gasoline per 9 miles run at 50 miles per 
hour, the horse power developed being 25. Find the heat unit consumption per Ihp. 
per minute and the thermal efficiency ; assuming gasoline to weigh 7 lb. per gallon. 

20. A two-cycle engine gives an indicator diagram in which the positive work 
area is 1000 ft. -lb., the negative work area 90 ft. -lb. The work at the brake is 700 
ft. -lb. Give two values for the mechanical efficiency. 

21. The engine in Problem 17 discharges 30 per cent of the heat it receives to the 
jacket. Find the water consumption in pounds per minute, if its initial temperature 
is 72° F. 

22. In Art. 344, what was the producer efficiency in the case of the Guldner en- 
gine, assuming its mechanical efficiency to have been 0.85 ? If the coal contained 
13,800 B. t. u. per pound, what was the coal consumption per brake hp.-hr. ? 

23. Given the indicator diagram of Fig. 158, plot accurately the TV diagram, the 
engine using 0.0462 lb. of substance per cycle. Draw the compressive path on the NT 
diagram by both of the methods of Art. 347. 

24. The engine in Problem 17 governs by throttling its charge. To what percent- 
age of the piston displacement should the clearance be decreased in order that the pres- 
sure after compression may be unchanged when the pre-compression pressure drops to 
10 lb. absolute ? What would be the object of such a change in clearance ? 

25. In the Diesel engine, Problem 7, by what percentages will the efficiency and 
capacity be affected, theoretically, if the supply of fuel, is cut off 50 per cent earlier in 

the stroke ? (i.e., cut-off occurs when the volume is — — — - + V a , Fig. 134.) 

26. For Martin's project (Art. 276), determine the velocity of the gas in the pipe 
line if it is transmitted 300 miles. Confirm, approximately, the estimate of power con- 
sumption, the plant operating continuously. If the coal contains 12,000 B. t. u. per 
pound and cost 50 cents per 2000 lb., and the gas contains 600 B. t. u. per cubic foot, 
what is the efficiency of the coal gas retorts ? 



198 



APPLIED THERMODYNAMICS 



300 


3 
















































240 






\ 
























\ 




















180 






\ 

< 


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120 


















































60 








" 


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' 
























r 


T^~*~^^ 


o^*~~*~— 































0.20 0.40 0.60 0.80 1.00 

Fig. 158. Prob. 23. — Indicator Diagram for Transfer. 



27. Under the conditions of Art. 335, develop a relation between piston displace- 
ment in cubic inches per minute; and Ihp., for four cylinder four-cycle single acting 
gasolene engines. Also find the relation between cylinder volume and Ihp. if engines 
run at 1500 r. p. m., and the relation between cylinder diameter and Ihp. if bore = stroke, 
at 1500 r. p. m. 

28. In an Otto engine, the range of pressures during compression is from 13 to 



130 lb., the compression curve pv^ 



Find the percentage of clearance. 



CHAPTER XII 
THEORY OF VAPORS 

354. Boiling of Water. If we apply heat to a vessel of water open 
to the atmosphere, an increase of temperature and a slight increase 
of volume may be observed. The increase of temperature is a gain 
of internal energy; the slight increase of volume against the constant 
resisting pressure of the atmosphere represents the performance of 
external work, the amount of which may be readily computed. After 
this operation has continued for some time, a temperature of 212° F. 
is attained, and steam begins to form. The 'water now gradually 
disappears. The steam occupies a much larger space than the water 
from which it was formed ; a considerable amount of external work is 
done in thus augmenting the volume against atmospheric pressure ; 
and the common temperature of the steam and the water remains con- 
stant at 212° F. during evaporation. 

355. Evaporation under Pressure. The same operation may be 
performed in a closed vessel, in which a pressure either greater or less 
than that of the atmosphere may be maintained. The water will now 
boil at some other temperature than 212° F. ; at a lower temperature, 
if the pressure is less than atmospheric, and at a higher temperature, if 
greater. The latter is the condition in an ordinary steam boiler. If 
the water be heated until it is all boiled into steam, it will then be 
possible to indefinitely increase the temperature of the steam, a result 
not possible as long as any liquid is present. The temperature at 
which boiling occurs may range from 32° F. for a pressure of 
0.089 lb. per square inch, absolute, to 428° F. for a pressure 
of 336 lb. ; but for each pressure there is a fixed temperature of 
ebullition. 

356. Saturated Vapor. Any vapor in contact with its liquid and 
in thermal equilibrium (i.e. not constrained to receive or reject heat) 

199 



200 APPLIED THERMODYNAMICS 

is called a saturated vapor. It is at the minimum temperature (that 
of the liquid) which is possible at the existing pressure. Its density 
is consequently the maximum possible at that pressure. Should it 
be deprived of heat, it cannot fall in temperature until after it has 
been first completely liquefied. If its pressure is fixed, its temperature 
and density are also fixed. Saturated vapor is then briefly definable 
as vapor at the minimum temperature or maximum density possible 
under the imposed pressure. 

357. Superheated Vapor. A saturated vapor subjected to ad- 
ditional heat at constant pressure, if in the presence of its liquid, 
cannot rise in temperature ; the only result is that more of the liquid 
is evaporated. When all of the liquid has been evaporated, or if the 
vapor is conducted to a separate vessel where it may be heated while 
not in contact with the liquid, its temperature may be made to rise, 
and it becomes a superheated vapor. It may be now regarded as an 
imperfect gas; as its temperature increases, it constantly becomes 
more nearly perfect. Its temperature is always greater, and its 
density less, than those properties of saturated vapor at the same 
pressure ; either temperature or density may, however, be varied at 
will, excluding this limit, the pressure remaining constant. At 
constant pressure, the temperature of steam separated from water 
increases as heat is supplied. 

The characteristic equation, P V = R T, of a perfect gas is inapplicable to steam. 
(See Art. 390.) The relation of pressure, volume, and temperature is given by 
various empirical formulas, including those of Joule (1), Rankine (2), Hirn (3), 
Racknel (4), Clausius (5), Zeuner (6), and Knoblauch Linde and Jakob (7). 
These are in some cases applicable to either saturated or superheated steam. 

Saturated Steam 

358. Thermodynamics of Vapors. The remainder of this text is 
chiefly concerned with the phenomena of vapors and their application 
in vapor engines and refrigerating machines. The behavior of vapors 
during heat changes is more complex than that of perfect gases. 
The temperature of boiling is different for different vapors, even at 
the same pressure ; but the following laws hold for all other vapors 
as well as for that of water : 



FORMATION OF STEAM 201 

(1) The temperatures of the liquid and of the vapor in contact with 

it are the same ; 

(2) The temperature of a specific saturated vapor at a specified pres- 

sure is always the same ; 

(3) The temperature and the density of a vapor remain constant 

during its formation from liquid at constant pressure ; 

(4) Increase of pressure increases the temperature and the density of 

the vapor ; * 

(5) Decrease of pressure lowers the temperature and the density ; 

(6) The temperature can be increased and the density can be decreased 

at will, at constant pressure, when the vapor is not in contact 
with its liquid ; 

(7) If the pressure upon a saturated vapor be increased without allow- 

ing its temperature to rise, the vapor must condense ; it cannot 
exist at the increased pressure as vapor (Art. 356). If the 
pressure is lowered while the temperature remains constant, the 
vapor becomes superheated. 

359. Effects of Heat in the Formation of Steam. Starting with 
a pound of water at 32° F., as a convenient reference point, the heat 
expended during the formation of saturated steam at any temperature 
and pressure is utilized in the following ways : 

(1) h units in the elevation of the temperature of the water. If the 
specific heat of water be unity, and t be the boiling point, 
h = t— 32 ; actually, h always slightly exceeds this, but the 
excess is ordinarily small, f J 

* Since mercury boils, at atmospheric pressure, at 675° F., common thermometers 
cannot be used for measuring temperatures higher than this ; but by filling the space in 
the thermometric £ube above the mercury with gas at high pressure, the boiling point 
of the mercury may be so elevated as to permit of its use for measuring flue gas 
temperatures exceeding 800° F. 

t According to Barnes' experiments (8), the specific heat of water decreases from 
1.0094 at 32° F. to 0.99735 at 100° F., and then steadily increases to 1.0476 at 428° F. 

$ In precise physical experimentation, it is necessary to distinguish between the 
value of h measured above 32° F. and atmospheric pressure, and that measured above 
32° F. and the corresponding pressure of the saturated vapor. This distinction is of no 
consequence in ordinary engineering work. 



202 APPLIED THERMODYNAMICS 

p( V — 'v\ 

(2) ry ^ — ' units in the expansion of the water (external work), p 

being the pressure per square foot and v and Fthe initial and 
final specific volumes of the water respectively. This quantity 
is included in item h, as above defined ; it is so small as to be 
usually negligible, and the total heat required to bring the 
water up to the boiling point is regarded as an internal energy 
change. 

(3) e = ry — J - units to perform the external work of increasing 

the volume at the boiling point from that of the water to that of 
the steam, If being the specific volume of the steam. 

(4) r units to perform the disgregation work of this change of state 
(Art. 15) ; items (3) and (4) being often classed together us L. 

The total heat expended per pound is then 
H= h + L = h + r + e. 

The values of these quantities vary widely with different vapors, even when 
at the same temperature and pressure ; in general, as the pressure increases, h 
increases and L decreases. Watt was led to believe (erroneously) that the sum of 
h and L for steam was a constant; a result once described as expressing "Watt's 
Law." This sum is now known to slowly increase with increase of pressure. 

360. Properties of Saturated Steam. It has been found experimentally 
that as p, the pressure, increases, t, h, e, and H increase, while r and L 
decrease. These various quantities are tabulated in what is known as a 
steam table.* A convenient form of table for quick reference is that in 

* Regnault's experiments were the foundation of the steam tables of Rankine (9), 
Zeuner (10), and Porter (11). The last named have been regarded as extremely accurate, 
and were adopted as standard for use in reporting trials of steam boilers and pumping 
engines by the American Society of Mechanical Engineers. They do not give all of the 
thermal properties, however, and have therefore been unsatisfactory for some purposes. 
The tables of Dwelshauevers-Dery (12) were based on Zeuner's; BuePs tables, origi- 
nally published in Weisbach's 3Iechanics (13), on Eankine's. The table in Kent's 
Mechanical Engineers' Pocket-Book is derived from Dwelshauevers-Dery and Buel. 
Peabody's tables are computed directly from Regnault's work (14). The principal 
differences in these tables were due to some uncertainty as to the specific volume of 
steam (15). The precise work of Holborn and Henning (16) on the pressure-tempera- 
ture relation and the adaptation by Davis (17) of recent experiments on the specific 
heat of superheated steam to the determination of the total heat of saturated steam (Art. 
388) have suggested the possibility of steam tables of greater accuracy. The most 
recent and satisfactory of these is that of Marks and Davis (18) , values from which 
are adopted in the remainder of the present text. (See pp. 247, 248.) 



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204 APPLIED THERMODYNAMICS 

which the values are plotted as coordinates, as in Fig. 159. Using this, 
we may find and check numerical values for the items in Art. 359, remem- 
bering that V= 0.017, as follows: 

p = 14.697 (lb. per sq. in.). p = 100.58 (lb. per sq. in.). 

t = 212. * = 328. 

h = 180.6. h = 298.4. 

H= 1146.6. £"=1182.0. 

L = 965.8. L= 883.6. 

r = 893.5. r = 802.4. 

e = 72.3. e = 81.2. 

Our original knowledge of these values was derived from the compre- 
hensive experiments of Regnault, whose empirical formula for the total 
heat of saturated steam was H= 1081.94 -+- 0.305 t. The recent investi- 
gations of Davis (17) show, however, that a more accurate expression is 

H = 1150.3 4- 0.3745 (t - 212) - 0.00055 (t - 212) 2 (Art. 388). 

(The total heat at 212° F. is represented by the value 1150.3.) Barnes' 
and other determinations of the specific heat of water permit of the com- 
putation of h ; and L = H—h. The value of e may be directly calculated 
if the volume W is known, and r= L — e. An inspection of Fig. 159 shows 
that the value of r has a straight line relation, approximately, with the 
temperature. This may be expressed by the formula r = 1061.3 — 0.79 f F. 
The method of deriving the steam volume, always tabulated with these 
other thermal properties, will be considered later. When saturated steam 
is condensed, all of the heat quantities mentioned are emitted in the 
reverse order, so to speak. Regnault's experiments were in fact made, 
not by measuring the heat absorbed during evaporation, but that emitted 
during condensation. Items h and r are both internal energy effects ; 
they are sometimes grouped together and indicated by the symbol E; 
whence H=E-{-e. The change of a liquid to its vapor furnishes the 
best possible example of what is meant by disgregation work. If there is 
any difficulty in conceiving what such work is, one has but to compare the 
numerical values of L and r for a given pressure. What becomes of the 
difference between L and e? The quantity L is often called the latent 
heat, or, more correctly, the latent heat of evaporation. The " heat in the 
water " referred to in the steam tables is h ; the " heat in the steam " is 
H, also called the total heat. 

361. Factor of Evaporation. In order to compare the total expen- 
ditures of heat for producing saturated steam under unlike condi- 
tions, we must know the temperature T, other than 32° F. (Art. 
359), at which the water is received, and the pressure p at which 



PRESSURE-TEMPERATURE 205 

steam is formed ; for as T increases, h decreases ; and as p increases, 
H increases. This is of much importance in comparing the results 
of steam boiler trials. At 14.7 lb. (atmospheric) pressure, for ex- 
ample, with water initially at the boiling point, 212° F., h = and 
H= L = 970.4 (from the table, p. 247). These are the conditions 
adopted as standard, and with which actual evaporative performances 
are compared. Evaporation under these conditions is described as 
being 

From (a feed water temperature of) and at (a pressure correspond- 
ing to the temperature of) 212° F. 

Thus, for p = 200, we find L = 843.2 and h = 354.9 ; and if the tem- 
perature of the water is initially 190° F., corresponding to the heat 
contents of 157.9 B. t. u., 

H=L + (354.9 - 157.9) = 843.2 + 197 = 1040.2. 

The ratio of the total heat actually utilized for evaporation to that 
necessary " from and at 212° F." is called the factor of evaporation. 

In this instance, it has the value 1040.2 -r- 970.4 = 1.07. Generally, 
if X, h refer to the assigned pressure, and A is the heat correspond- 
ing to the assigned temperature of the feed water, then the factor of 

evaporation is 

F=[L+ (A-A )]-*- 970.4. 

362. Pressure-temperature Relation. Regnault gave, as the result of his ex- 
haustive experiments, thirteen temperatures corresponding to known pressures 
at saturation. These range from — 32° C. to 220° C. He expressed the relation 
by four formulas (Art. 19) ; and no less than fifty formulas have since been 
devised, representing more or less accurately the same experiments. The deter- 
minations made by Holborn and Henning (16) agree closely with those of Reg- 
nault; as do those by Wiebe (19) and Thiesen and Scheel (20) at temperatures 
below the atmospheric boiling point. There is no satisfactory data at tempera- 
tures exceeding 500° F. 

The steam table shows that, beginning at 32° F., the pressure rises with the 
temperature, at first slowly and afterward much more rapidly. It is for this 
reason that two pressure-temperature curves, with different pressures scales, have 
been used in Fig. 159,. the low-scale curve being used for low pressures. If the 
high-scale curve were extended downward, It would be difficult to ascertain accu- 
rately the pressure changes below atmospheric for small differences of tempera- 
ture. The fact that slight increases of temperature accompany large increases of 
pressure in the working part of the range seems fatal to the development of the 
engine using saturated steam, the high temperature of heat absorption shown by 



206 APPLIED THERMODYNAMICS 

Carnot to be essential to efficiency being unattainable without the use of pressures 
mechanically objectionable. 

A recent formula for the relation between pressure and temperature is (Power, 
« 8, 1910) < = 20M-101, 

in which t is* the Fahrenheit temperature and p the pressure in pounds per square 
inch. 

363. Pressure and Volume. Fairbairn and Tate ascertained experimentally 
in 1860 the relation between pressure and volume at a few points; some experi- 
ments were made by Hirn; and Battelli has reported results which have been 
examined by Tumlirz (21). More recent experiments by Knoblauch, Linde, and 
Klebe (1905) (22) give the formula 

pv = 0.5962 T-p(l + 0.0014 p) ( 150 ' 30( ^ 000 - 0.0833Y 

in which p is in pounds per square inch, v in cubic feet per pound, and Tin degrees 
absolute. This may be compared with Wood's formula (23), 

pv = 0.6732 T - 1?M . 

r yO.22 

• 1 3_ 

A simple empirical formula is that of Rankine, Pfie == constant, or that of Zeuner, 
PJ71.0646 _ constant. These forms of expression must not be confused with the 
PV n = c equation for various poly tropic paths. An indirect method of determin- 
ing the volume of saturated steam is to observe the value of some thermal prop- 
erty, like the latent heat, per pound and per cubic foot, at the same pressure. 

The incompleteness of experimental determinations, with, the diffi- 
culty in all cases of ensuring experimental accuracy, have led to the use of 
analytical methods (Art. 368) for computing the specific volume. The 
values obtained agree closely with those of Knoblauch, Linde, and Klebe. 

364. Wet Steam. Even when saturated steam is separated from 
the mass of water from which it has been produced, it nearly always 
contains traces of water in suspension. The presence of this water 
produces what is described as wet steam, the wetness being an indi- 
cation of incomplete evaporation. Superheated steam, of course, 
cannot be wet. Wet steam is still saturated steam (Art. 356); the 
temperature and density of the steam are not affected by the pres- 
ence of water. 

The suspended water must be at the same temperature as the 
steam ; it therefore contains, per pound, adopting the symbols of 
Art. 359, h units of heat. In the total mixture of steam and water, 
then, the proportion of steam being x, we write for Z, xL ; for r, xr ; 
for e, xe ; for E, xr + h ; while, h remaining unchanged, H =h -f xL. 



FORMATION OF STEAM 



207 




o 

Fig. 160. Arts. 365, 366, 379. 
of Steam Formation. 



Paths 



The factor of evaporation (Art. 361), wetness considered, must be 
correspondingly reduced ; it is F = \xL + (h — A )] -r- 970.4. 

The specific volume of wet steam is W w = V+x ( W— V) = xZ+ F, 
where Z= W- V. For dry steam, x=l, and W w = V+ ( W- V) = W. 
The error involved in assuming W w = xWis usually inconsiderable, 
since the value of V is comparatively small. 

365. Limits of Existence of Saturated Steam. In Fig. 160, let 
ordinates represent temperatures, and abscissas, volumes. Then ah 
is a line representing possible condi- t 
tions of water as to these two proper- 
ties, which may be readily plotted if 
the specific volumes at various tem- 
peratures are known ; and cd is a 
similar line for steam, plotted from the 
values of IT and t in the steam table. 
The lines ah and cd show a tendency 
to meet (Art. 379). The curve cd is 
called the curve of saturation, or of con- 
stant steam weight ; it represents all possible conditions of constant 
weight of steam, remaining saturated. It is not a path, although 
the line ah is (Art. 363). States along ah are those of liquid; the 
area bade includes all wet saturated states ; along dc, the steam is 
dry and saturated; to the right of dc, areas include superheated 
states. 

366. Path during Evaporation. Starting at 32°, the path of the 
substance daring heating and evaporation at constant pressure would 
be any of a series of lines aef, ahi, etc. The curve ah is sometimes 
called the locus of boiling points. If superheating at constant pres- 
sure occur after evaporation, then (assuming Charles' law to hold) 
the paths will continue as fg, ij, straight lines converging at 0. 
For a saturated vapor, wet or dry, the isothermal can only be a straight 
line of constant pressure. 

367. Entropy Diagram. Figure 161 reproduces Fig. 160 on the 
entropy plane. The line ah represents the heating of the water at 
constant pressure. Since the specific heat is slightly variable, the 



208 



APPLIED THERMODYNAMICS 



increase of entropy must be computed for small differences of tem- 
perature. The more complete steam tables give the entropy at various 
boiling points, measured above 32°. Let evaporation occur when the 




D g M v, p t 

Fig. 161. Arts. 367, 369-373, 376, 379, 386, 426. — The Steam Dome. 

temperature is T b . The increase of entropy from the point b (since 
the temperature is constant during the formation of steam at constant 
pressure) is simply L -f- (T h + 459.6), which is laid off as be. Other 
points being similarly obtained, the saturation curve ed is drawn. 
The paths from liquid at 32° to dry saturated steam are abc, a VN, 
aUS, etc. 

The factor of evaporation may be readily illustrated. Let the area 
eUSf represent i a2 , the heat necessary to evaporate one pound from and 
at 212° F. The area gjbch represents the heat necessary to evaporate one 
pound at a pressure b from a feed-water temperature j. The factor of 
evaporation is gjbch -*- e USf For wet steam at the pressure b, it is, for 
example, gjbik -s- eUSf. 

368. Specific Volumes: Analytical Method. This was developed by 
Clapeyron in 1834. In Fig. 162, let abed represent a Carnot cycle in 
which steam is the working substance and the range of temperatures is. 
dT. Let the substance be liquid along da and dry saturated vapor along be. 



VOLUME OF VAPOR 



209 



The heat area abfe is L; the work area abed is (L •+■ T)dT. In Fig. 163, 
let abed represent the corresponding work area on the pv diagram. Since 
the range of temperatures is only dT, the range of pressures may be 



Figs. 162 and 163. Arts. 368, 406, 603. — Specific Volumes by Clapeyron's Method. 

taken as dP; whence the area abed in Fig. 163 is dP(W — V), where W 
is the volume along be, and Fthat along ad. This area must by the first 
law of thermodynamics equal (778 L -+- T)dT\ whence 

W _ yjn*lL . *1 and W= V+^Ml. 
T dP T • dP 

Thus, if we know the specific volume of the liquid, and the latent heat 
of vaporization, at a given temperature, we have only to determine the 

dT 

differential coefficient — in order to compute the specific volume of the 

vapor. The value of this coefficient may be approximately estimated from 
the steam table; or may be accurately ascertained when any correct formula 
for relation between P and T is given. The advantage of this indirect 
method for ascertaining specific volumes arises from the accuracy of 
experimental determinations of T, L, and P. 

369. Entropy Lines. In Fig. 161, let ab be the water line, cd 
the saturation curve ; then since the horizontal distance between 
these lines at any absolute temperature T is equal to L-t- T, we 
deduce that, for steam only partially dry, the gain of heat in passing 
from the water line toward cd being xL instead of L, the gain of 
entropy is xL -r- T instead of L -s- T. If on be and ad we lay off hi 
and al = x • be and x ■ ad, respectively, we have two points on the 
constant dryness curve il, along which the proportion of dryness is x. 
Additional points will fully determine the curve. The additional 
curves «w, pq, etc., are similarly plotted for various/ values of x, all 
of the horizontal intercepts between ab and cd being divided in the 
same proportions by any one of these curves. 



210 APPLIED THERMODYNAMICS 

370. Constant Heat Curves. Let the total heat at o be R. To 
find the state at the temperature be, at which the total heat may also 
equal H, Ave remember that for wet steam JT= h + xL, whence 
x = QH— K) -s- L = bp -r- be. Additional points thus determined for 
this and other assigned values of H give the constant total heat 
curves op, mr, etc. The total heat of saturated vapor is not, however, 
a cardinal property (Art. 10). The state points on this diagram 
determine the heat contents only on the assumption that heat has 
been absorbed at constant pressure ; along such paths as abc, aUS t 
aVN, etc. 

371. Negative Specific Heat. If steam passes from o to r, Fig. 161, 
heat is absorbed (area sort) while the temperature decreases. Since the satu- 
ration curve slopes constantly downward toward the right, the specific heat 
of steam kept saturated is therefore negative. The specific heat of a vapor 
can be positive only when the saturation curve slopes downward to the left, 
like cu, as in the case, for example, of the vapor of ether (Fig. 315). The 
conclusion that the specific heat of saturated steam is negative was 
reached independently by Rankine and Clausius in 1850. It was experi- 
mentally verified by Hirn in 1862 and by Cazin in 1866 (24). The 
physical significance is simply that when the temperature of dry saturated 
steam is increased adiabatically, it becomes superheated; heat must be 
abstracted to keep it saturated. On the other hand, when dry saturated 
steam expands, the temperature falling, it tends to condense, and 
heat must be supplied to keep* it dry. If steam at c, Fig. 161, having 
been formed at constant pressure, works along the saturation curve to A 7 ", 
its heat contents are not the same as if it had been formed along a VN, 
but are greater, being greater also than the "heat contents" at c. 

372. Liquefaction during Expansion. If saturated steam expand adia- 
batically from c, Fig. 161, it will at v have become 10 per cent wet. If 
its temperature increase adiabatically from v, it will at c have become 
dry. If the adiabatic path then continue, the steam will become superheated. 
Generally speaking, liquefaction accompanies expansion and drying or 
superheating occurs during compression. If the steam is very wet to begin 
with, say at the state x, compression may, however, cause liquefaction, and 
expansion may lead to drying. Water expanding adiabatically (path bz) 
becomes partially vaporized. Vapors may be divided into two classes, 
depending upon whether they liquefy or dry during adiabatic expansion 
under ordinary conditions of initial dryness. At usual stages of dryness 
and temperature, steam liquefies during expansion, while ether becomes 
dryer, or superheated. 



INTERNAL ENERGY OF VAPOR 



211 



373. Inversion. Figure 161 shows that when x is about 0.5 the constant dry- 
ness lines change their direction of curvature, so that it is possible for a single 
adiabatic like DE to twice cut the same dryness curve ; x may therefore have the 
same value at the beginning and end of expansion, as at D and E. Further, it 
may be possible to draw an adiabatic which is tangent to the dryness curve at A. 
Adiabatic expansion below A tends to liquefy the steam ; above .4, it tends to dry 
it. During expansion along the dryness curve below A, the specific heat is nega- 
tive; above A, it is positive. By finding other points like A, as F, G, on similar 
constant dryness curves, a line BA may be drawn, which is called the zero line or 
line of inversion. During expansion along the dryness lines, the specific heat 
becomes zero at their intersection with AB, where they become tangent to the 
adiabatics. If the line AB be projected so as to meet the extended saturation 
curve dc, the point of intersection is the temperature of inversion. There is no 
temperature of inversion for dry steam (Art. 379), the saturation curve reaching 
an upper limit before attaining a vertical direction. 

374. Internal Energy. In Fig. 164, let 2 be the state point of a wet vapor. 
Lay off 2 4 vertically, equal to (T ~ L)(L-r). Then 1 2 4 3 (3 4 being drawn 
horizontally and 1 3 vertically) is equal to t 



12 x 24-^ 



(L — r) = x(L — r) . 



This quantity is equal to the external work of 
vaporization = xe, which is accordingly repre- 
sented by the area 12 4 3. The irregular 
area 6 5 13 4 7 then represents the addition 
of internal energy, 6 5 18 having been ex- 
pended in heating the water, and 8 3 4 7=xr 
being the disgregation work of vaporization. 




Fig. 



164. Art. 374. — Internal Energy 
and External Work. 



375. External Work. Let MN, Fig. 165, be any path in the saturated region. 
The heat absorbed is mMNn. Construct Mcba, Nfed, as in Art. 374. The inter- 
nal energy has increased from Oabcm to Odefn, the 
amount of increase being adefnmcb. This is greater 
than the amount of heat absorbed, by deiMcba — iNJ, 
which difference consequently measures the external 
work done upon the substance. Along some such curve 
as A Y, it will be found that external work has been 
done by the substance. 




o 
Fig. 165. Art. 375. —In 
ternal Energy of Steam. 



376. The Entropy Diagram as a Steam Table. In 
Fig. 161, let the state point be H. We have T= HI, 
from which P may be found. HJ is made equal to (T +- L)(L — r), whence 
Oa VKJI — E and VH.TK = xe. Also x = VH -=- VN, the entropy measured from 
the water line is VH, the momentary specific heat of the water along the dif- 
ferential path jL is gjLM + Tj', xL = PVHI, xr = KJIP, h = OaVP, and 
H = Oa VHI. The specific volume is still to be considered. 



212 



APPLIED THERMODYNAMICS 




Fig 



166. Art. 377. — Constant 
Volume Lines. 



377. Constant Volume Lines. In Fig. 166, let J A be the water 
line, BCr the saturation curve, and let vertical distances below ON 
represent specific volumes. Let xs equal the volume of boiling water, 

sensibly constant, and of comparatively 
small numerical value, giving the line ss. 
From any point B on the saturation 
curve, draw BD vertically, making CD 
represent by its length the specific volume 
at B. Draw BA horizontally, and AE 
vertically, and connect the points i^andi). 
Then ED shows the relation of volume of 
vapor and entropy of vapor, along AB, 
the two increasing in arithmetical ratio. 
Find the similar lines of relation KL and 
HF for the temperature lines JI and YGr. 
Draw the constant volume line TD, and 
find the points on the entropy plane 
w, v, B, corresponding to £, u, D. The line of constant volume wB 
may then be drawn, with similar lines for other specific volumes, qz, 
etc. The plotting of such lines on the entropy plane permits of the 
use of this diagram for obtaining 
specific volumes (see Fig. 175). 

378. Transfer of Vapor States. In 
Fig. 167, we have a single represen- 
tation of the four coordinate planes 
pt, tn, nv, and pv. Let ss be the line 
of water volumes, ah and ef the satura- 
tion curve, Cd the pressure-tempera- 
ture curve (Art. 362), and Op the 
water line. To transfer points a, b on 
the saturation curve from the pv to the 
tn plane, we have only to draw aC, 
Ce, bd, and df. To transfer points 
like i, Z, representing wet states, we 

first find the vn lines qh and rg as in Art. 377, and then project 
ij,jk, Im, and mn (25). 




Fig. 167. Art. 378. — Transfer of 
Vapor States. 



CRITICAL TEMPERATURE 213 

Consider any point t on the pv plane. By drawing tu and uv we 
find the vertical location of this point in the tn plane. Draw wA and 
xB, making zB equal to the specific volume of vapor at x (equal to 
BF on the pv plane). Draw AB and project t to c. Projecting this 
last point upward, we have D as the required point on the entropy 
plane. 

379. Critical Temperature. The water curve and the curve of saturation 
in Figs. 160 and 161 show a tendency to meet at their upper extremities. 
Assuming that they meet, what are the physical conditions at the critical 
temperature existing at the point of intersection f It is evident that here 
L = 0, r = 0, and e = 0. The substance would pass immediately from the 
liquid to the superheated condition ; there would be no intermediate state 
of saturation. No external work would be done during evaporation, and, 
conversely, no expenditure of external work could cause liquefaction. A 
vapor cannot be liquefied, when above its critical temperature, by any 
pressure whatsoever. The density of the liquid is here the same as that 
of the vapor: the two states cannot be distinguished. The pressure re- 
quired to liquefy a vapor increases as the critical temperature is approached 
(moving upward) (Arts. 358, 360) ; that necessary at the critical temperature 
is called the critical pressure. It is the vapor pressure corresponding to the 
temperature at that point. The volume at the intersection of the saturation 
curve and the liquid line is called the critical volume. The " specific heat 
of the liquid " at the critical temperature is infinity. 

The critical temperature of carbon dioxide is 88.5° F. This substance is 
sometimes used as the working fluid in refrigerating machines, particularly on 
shipboard. It cannot be used in the tropics, however, since the available supplies 
of cooling water have there a temperature of more than 88.5° F., making it im- 
possible to liquefy the vapor. The carbon dioxide contained in the microscopic 
cells of certain minerals, particularly the topaz, has been found to be in the critical 
condition, a line of demarcation being evident, when cooling was produced, and 
disappearing with violent frothing when the temperature again rose. Here the 
substance is under critical pressure; it necessarily condenses with lowering of 
temperature, but cannot remain condensed at temperatures above 88.5° F. Ave- 
narius has conducted experiments on a large scale with ether, carbon disulphide, 
chloride of carbon, and acetone, noting a peculiar coloration at the critical point (26). 

For steam, Regnault's formula for H (Art. 360), if we accept the approximation 
h = t - 32°, would give L = H — h = 1113.94 - 0.695 t, which becomes zero when 
t = 1603° F. Davis' formula (Art. 360) (likewise not intended to apply to temper- 
atures above about 400° F.) makes L = when t = 1709° F. The critical tempera- 
ture for steam has been experimentally ascertained to be actually much lower, the 
best value being about 689° F. (27). Many of the important vapors have been 
studied in this direction by Andrews. 



214 



APPLIED THERMODYNAMICS 



380. Physical States. We may now distinguish between the gaseous 
conditions, including the states of saturated vapor, superheated vapor, and 
true gas. A saturated vapor, which may be either dry or wet, is a gaseous 
substance at its maximum density for the given temperature or pressure; 
and below the critical temperature. A superheated vapor is a gaseous sub- 
stance at other than maximum density whose temperature is either less 
than, or does not greatly exceed, the critical temperature. At higher tempera- 
tures, the substance becomes a true gas. All imperfect gases may be regarded 
as superheated vapors. 

Air, one of the most nearly perfect gases, shows some deviations from Boyle's law 
at pressures not exceeding 25001b. per square inch. Other substances show far more 
marked deviations. In Fig. 168, QP is an equilateral hyperbola. The isothermals 

for air at various temperatures centi- 
grade are shown above. The lower 
curves are isothermals for carbon di- 
oxide, as determined by Andrews (28). 
They depart widely from the perfect 
gas isothermal, PQ. The dotted lines 
show the liquid curve and the satura- 
tion curve, running together at a, at the 
critical temperature. There is an evi- 
dent increase in the irregularity of the 
curves as they approach the critical tem- 
perature (from above) and pass below 
it. The curve for 21.5° C. is particu- 
larly interesting. From b to c it is a 
liquid curve, the volume remaining 
practically constant at constant temperature in spite of enormous changes of pres- 
sure. From b to d it is a nearly straight horizontal line, like that of any vapor 
between the liquid and the dry saturated states ; while from d to e it approaches 
the perfect gas form, the equilateral hyperbola. All of the isothermals change 
their direction abruptly whenever they ap- 
proach either of the limit curves af or ag. 

381. Other Paths of Steam Formation. 
The discussion has been limited to the 
formation of steam at constant pressure, 
the method of practice. Steam might con- 
ceivably be formed along any arbitrary 
path, as for instance in a closed vessel at 
constant volume, the pressure steadily in- 
creasing. Since the change of internal 
energy of a substance depends upon its 
initial and final states only, and not on the intervening path, a change of path 
affects the external work only. For formation at constant volume, the total heat 
equals E, no external work being done. If in Fig. 169 water at c could be com- 



100 
95H 



/ 




Fig. 168. Art. 380. — Critical Temperature. 




Fig. 169. Art. 381. — Evaporation at 
Constant Volume. 



SUPERHEATED STEAM 215 

pletely evaporated along en at constant volume, the area. aend would represent the 
addition of internal energy and the total heat received. If the process be at con- 
stant pressure, along cbn, the area acbnd represents the total heat received and the 
area cbn represents the external work done. 

382. Vapor Isodynamic. A saturated vapor contains heat above 32° F. equal 
to h + r + e; or, at some other state, to h 1 + i\ + e v If the two states are isody- 
namic (Art. 83), h + r = li x + r v a condition which is impossible if at both states 
the steam be dry. If the steam be wet at both states, h -j- xr = h x + x x r v Let p, 
p v v be given ; and let it be required to find v v the notation being as in Art. 364. 

H ' 1 J 7*T* • J) 

We have x 1 — —^ 1 , all of these quantities being known or readily ascertain- 

able. Then 

r i 
If x = 1.0, the steam being dry at one state, x x = — — 1 and 

Substitution of numerical values then shows that if p exceed pi, v is less than v\ ; 
i.e. the curve slopes upward to the left on the pv diagram : and x is less than 
x v The curve is less " steep " than the saturation curve. Steam cannot be worked 
isodynamically and remain dry ; each isodynamic curve meets the saturation curve 
at a single point. 

Superheated Steam 

383. Properties : Specific Heat. In comparatively recent years, superheated 
steam has become of engineering importance in application to reciprocating en- 
gines and turbines and in locomotive practice. 

Since superheated steam exists at a temperature exceeding that of saturation, 
it is important to know the specific heat for the range of superheating. The first 
determination was by Regnault (1862), who obtained as mean values k = 0.4805, 
I = 0.346, y = 1.39. Fenner found / to be variable, ranging from 0.341 to 0.351. 
Hirn, at a later date, concluded that its value must vary with the temperature. 
Weyrauch (29), who devoted himself to this subject from 1876 to 1904, finally 
concluded that the value of k increased both with the pressure and with the 
amount of superheating (range of temperature above saturation), basing this con- 
clusion on his own observations as collated with those of Regnault, Hirn, Zeuner, 
Mallard and Le Chatelier, Sarrau and Veille, and Langen. Rankine presented a 
demonstration (now admitted to be fallacious) that the total heat of superheated 
steam was independent of the pressure. At very high temperatures, the values 
obtained by Mallard and Le Chatelier in 1883 have been generally accepted by 
metallurgists, but they, do not apply at temperatures attained in power engineer- 
ing. A list by Dodge (30) of nineteen experimental studies on the subject shows 
a fairly close agreement with Regnault's value for k at atmospheric pressure and 
approximately 212° F. Most experimenters have agreed that the value increases 
with the pressure, but the law of variation with the temperature has been in 



216 



APPLIED THERMODYNAMICS 



doubt. Holborn's results (31) as expressed by Kutzbach (32) would, if the em- 
pirical formula held, make k increase with the temperature up to a certain limit, 
and then decrease, apparently to zero. 

384. Knoblauch and Jakob Experiments. These determinations (33) 
have attracted much attention. They were made by electrically super- 























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TEMPERATURE- DEG.REES FAHRENHEIT 



Fig. 170. Arts. 384, 421. — Specific Heat of Superheated Steam. Knoblauch and 

Jakob Results. 

heating the steam and measuring the input of electrical energy, which 
was afterward computed in terms of its heat equivalent. These experi- 
menters found that k increased with the pressure, and (in general) 



SPECIFIC HEAT 



217 



decreased with the temperature up to a certain point, afterward increas- 
ing (a result the reverse in this respect of that reported by Holborn). 
Figure 170 shows the results graphically. Greene (34) has used these in 
plotting the lines of entropy of superheat, as described in Art. 398. The 
Knoblauch and Jakob values are more widely used than any others experi- 
mentally obtained. 

385. Thomas' Experiments. In these, the electrical method of heating 
and a careful system of radiation corrections were employed (35). The 
conclusion reached was that k increases with increase of pressure and 
decreases with increase of temperature. The variations are greatest near 
the saturation curve. The values given included pressures from 7 to 500 lb. 





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Fig. 171. Arts. 385, 388, 398, 417, Prob. 42. — Specific Heat of Superheated Steam. 
Thomas' Experiments. 



per square inch absolute, and superheating ranging up to 270° F. The 
entropy lines and total heat lines are charted in Thomas' report. Within 
rather narrow limits, the agreement is close between these and the Knob- 
lauch and Jakob experiments. The reasons for disagreement outside 
these limits have been scrutinized by Heck (36), who has presented a 
table of the properties of superheated steam, based on these and other data. 
The experiments may be so readily duplicated that there is every reason 
for deferring final tabulating until a full set of confirmatory values shall 
have been obtained. Figure 171 shows the Thomas results graphically. 

386. Total Heat. As superheated steam is almost invariably formed 
at constant pressure, the path of formation resembles abcW, Fig. 161, ab 



218 APPLIED THERMODYNAMICS 

being the water line and cd the saturation curve. Its total heat is then 
H c -\-k(T—t), where T, t refer to the temperatures at IF and c. If we 
take Eegnault's value for #,,1081.94 + 0.305* (Art. 360), then, using 
k = 0.4805, we find the total heat of superheated steam to be 1081.94 — 
0.1755^ + 0.4805 T. A purely empirical formula, in which Pis the pres- 
sure in pounds per square foot, is H= 0.4805( T- 10.37 P - 25 ) +857.2. 
For accurate calculations, the total heat must be obtained by using correct 
mean values for k during successive short intervals of temperature between 
t and T. 

387. Variations of k. Dodge (37) has pointed out a satisfactory method 
for computing the law of variation of the specific heat. Steam is passed 
through a small orifice so as to produce a constant reduction in a constant 
pressure. It is superheated on both sides the orifice ; but, the heat con- 
tents remaining constant during the throttling operation, the temperature 
changes. Let the initial pressure be p, the final pressure p v Let one 
observation give for an initial temperature t, a final temperature ^; and 
let a second observation give for an initial temperature T, a final tempera- 
ture T Y . Let the corresponding total heat contents be h, h x , H, H x . Then 
h-H=k p (t- T) and \ - H 1 = k p (t x - 7\): But h = h 1} H= H\, whence 

~fc I rp 

li — H=h 1 — H x and - p - = • If we know the mean value of k for any 

k p t — T 

given range of temperature, we may then ascertain the mean value for a 

series of ranges at various pressures. 

388. Davis' Computation of H* The customary method of deter- 
mining k has been by measuring the amount of heat necessarily added 
to saturated steam in order to produce an observed increase of tem- 
perature. Unfortunately, the value of H for saturated steam has 
not been known with satisfactory accuracy ; it is therefore inade- 
quate to measure the total heat in superheated steam for comparison 
with that in saturated steam at the same pressure. Davis has shown 
(17) that since slight errors in the value of H lead to large errors 
in that of k, the reverse computation — using known values of k to 
determine H — must be extremely accurate ; so far so, that while 
additional determinations of the specific heat are in themselves to be 
desired, such determinations cannot be expected to seriously modify 
values of H as now computed. 

The basis of the computation is, as in Art. 387, the expansion of 
superheated steam through a non-conducting nozzle, with reduction 



VALUE OF H 219 

of temperature. Assume, for example, that steam at 38 lb. pres- 
sure and 300° F. expands to atmospheric pressure, the temperature 
becoming 286° F. The total heat before throttling we may call 
H c — H b + Jc^Tc — T b ~), in which H b is the total heat of saturated 
steam at 38 lb. pressure, T c = 300° F., and T b is the temperature of 
saturated steam at 38 lb. pressure, or 264.2° F. After throttling, 
similarly, H d = H e -f Jc 2 (T d — T e ), in which H e is the total heat of 
saturated steam at atmospheric pressure, T e is its temperature 
(212° F.), and T d is 286° F. Now H d = H c , and H e = 1150.4 ; while 
from Fig. 171 we find k x = 0.57 and Jc 2 = 0.52 ; whence 

H b = - 0.57(300 - 264.2) + 1150.4 + 0.52(286 - 212)= 1168.47. 

The formula given by Davis as a result of the study of various 
throttling experiments may be found in Art. 360. The total heat 
of saturated steam at some one pressure (e.g. atmospheric) must be 
known . 

A simple formula (that of Smith), which expresses the Davis results with an 
accuracy of 1 per cent, between 70° and 500°, was given in Power, February 8, 1910. 

Itis H = 1826 + t- 1 ^ 000 , 

1620 - t 

t being the Fahrenheit temperature. 

389. Factor of Evaporation. The computation of factors of evapora- 
tion must often include the effect of superheat. The total heat of super- 
heated steam — which we may call H s — may be obtained by one of the 
methods described in Art. 386. If 7i is the heat in the water as sup- 
plied, the heat expended is H s — h and the factor of evaporation is 

(H s -h ) + 970.4. 

390. Characteristic Equation. Zeuner derives as a working formula, 
agreeing with Hirn's experiments on specific volume (38), 

PV= 0.64901 T- 22.5819 P 025 , 

in which P is in pounds per square inch, V in cubic feet per pound, and 
T in degrees absolute Fahrenheit. This applies closely to saturated as 
well as to superheated steam, if dry. Using the same notation, Tumlirz 
gives (39) from Battelli's experiments, 

PV= 0.594 T- 0.00178 P. 

The formulas of Knoblauch, Linde and Jakob, and of Wood, both given 
in Art. 363, may also be applied to superheated steam, if not too highly 



220 APPLIED THERMODYNAMICS 

superheated. At very high temperatures, steam behaves like a perfect gas, 
following closely the law PV= RT. Since the values of R for gases are 
inversely proportional to their densities, we find R for steam to be 85.8. 

391. Adiabatic Equation. Using the value just obtained for R, and Regnault' s 
constant value 0.4805 for k, we find y = 1.298. The equation of the adiabatic 
would then be joy 1 - 298 = c. This, like the characteristic equation, does not hold 
for wide state ranges ; a more satisfactory equation remains to be developed 
(Art. 397). The exponential form of expression gives merely an approximation 
to the actual curve. 

Paths of Vapors 

392. Vapor Adiabatics. It is obvious from Art. 372 that during 
adiabatic expansion of a saturated vapor, the condition of dryness 

must change. We now compute the equa- 
tion of the adiabatic for any vapor. In 
Fig. 172, consider expansion from b to e. 
Draw the isothermals T, t. We have 



n n b 



= C^+ L± and n c -n d = x -^, T x be- 



Fig. 172. Art. 392.— Equa- ing the variable temperature along da. But 

tion of Vapor Adiabatic. i -_e j_i -a i i. £ j/l t -j u 

n b = n c , and it the specific heat of the liquid be 
constant and equal to c, ^k= elog e - -f- =^, the desired equation. 

V V J. 

If the vapor be only X dry at 6, then 

-2-2 = C log e - + . 

393. Applications. This equation may of course be used to derive the results 
shown graphically in Art. 373. For example, for steam initially dry, we may 
make X = 1, and it will be always found that x c is less than 1. To show that 
water expanding adiabatically partially vaporizes, we make X = 0. To determine 
the condition under which the dryness may be the same after expansion as before 
it, we make x = X. 

394. Approximate Formulas. Rankine found that the adiabatic might be 
represented approximately by the expression, 

io 

?F 9 = constant; 

which holds fairly well for limited ranges of pressure when the initial dryness is 
1.0, but which gives a curve lying decidedly outside the true adiabatic for any con- 
siderable pressure change. The error is reduced as the dryness decreases, down to 
a certain limit. Zeuner found that an exponential equation might be written in 



STEAM ADIABATICS 



221 



the form PV n = constant, if the value of n were made to depend upon the initial 
dryness. He represented this by 

n = 1.035 + 0.100 X, 

for values of X ranging from 0.70 to 1.00, and found it to lead to sufficiently accu- 
rate results for all usual expansions. For a compression from an initial dryness x, 
n = 1.034 + 0.11 x. Where the steam is initially dry, n = 1.135 for expansion and 
1.144 for compression. There is seldom any good reason for the use of exponential 
formulas for steam adiabatics. The relation between the true adiabatic and that 
described by the exponential equation is shown by the curves of Fig. 173 after 




Fig. 173. Arts. 394, 395. 



10 15 

Adiabatic and Saturation Curves. 



Heck (40). In each of these five sets of curves, the solid line represents the 
adiabatic, while the short-dotted lines are plotted from Zeuner's equation, and the 
long-dotted lines represent the constant dryness curves. In I and IT, the two 
adiabatics apparently exactly coincide, the values of x being 1.00 and 0.75. In 
III, IV, and V, there is an increasing divergence, for x = 0.50, 0.25 and 0. Case 
V is for the liquid, to which no such formula as those discussed could be expected 
to apply. 



395. Adiabatics and Constant Dryness Curves. The constant dryness curves 
I and II in Fig. 173 fall above the adiabatic, indicating that heat is absorbed during 
expansion along the constant dryness line. Since the temperature falls during 
expansion, the specific heat along these constant dryness curves, within the limits 
shown, must necessarily be negative, a result otherwise derived in Art. 373. The 
points of tangency of these curves with the corresponding adiabatics give the 
points of inversion, at which the specific heat changes sign. 



222 



APPLIED THERMODYNAMICS 



396. External Work 
P J^to pv, assuming pv" 



The work during adiabatic expansion from 
P V n , is represented by the formula 

PV-pv 
n-1 

More accurately, remembering that the work done equals the loss of 
internal energy, we find its value to be H— h + XR — zr, in which 
H and h denote the initial and final heats of the liquid. 



397. Superheated Adiabatic. Three cases are suggested in Fig. 174, paths jm, 
jk, de, the initially superheated vapor being either dry, wet, or superheated at the 





T 


h 






/ - 


V 




A 




' m 











Fig. 171. Art. 397. — Steam Adiabatics. 



end of expansion. If k be the mean value of the specific heat of superheated 
steam for the range of temperatures in each case, then 



for 



> m > c log« -^7 
± 1 






k log, 



forjfc, c log e p + ff + k loge Ti = *£k. 

± a + b ±b 1 a 



for de, c log e — b - 

1 9 



J-b J-b 



Lf 



+ h 



T, 



398. Entropy Lines for Superheat. Many problems in superheated 
steam are conveniently solved by the use of a carefully plotted entropy 
diagram, as shown in Fig. 175.* The plotting of the curves within the 
saturated limits has already been explained. At the upper right-hand 
corner of the diagram there appear constant pressure lines and constant 
total heat curves. The former may be plotted when we know the mean 
specific heat k at a stated pressure between the temperatures T and t : the 

T 

entropy gained being k log e — . The lines of total heat are determined 



* This diagram is based on saturated steam tables embodying Regnault's results, and 
on Thomas' values for k; it does not agree with the tables given on pages 247, 248. The 
same remark applies to Figs. 159 and 177. 



STEAM ENTROPY DIAGRAM 



223 




Fig. 175. Arts. 377, 398, 401, 411, 417, 516, Problems. 



224 



APPLIED THERMODYNAMICS 



by the following method: — For saturated steam at 103.38 lb. pressure, 
H= 1182.6, r=330° F. As an approximation, the total heat of 1200 
B. t. u. will require (1200 - 1182.6)-=- 0.4805 = 36.1° F. of superheating. 
For this amount of superheating at 100 lb. pressure, the mean specific 
heat is, according to Thomas (Fig. 171), 0.604; whence the rise in tem- 
perature is 17.4 -j- 0.604 = 28.7° F. For this range (second approxima- 
tion), the mean specific heat is 0.612, whence the actual rise of temperature 
is 17.4 -s- 0.612 = 28.4° F. No further approximation is necessary ; the 
amount of superheating at 1200 B. t. u. total heat may be taken as 28° F., 

which is laid off 
vertically from the 
point where the satu- 
ration curve crosses 
the line of 330° F., 
giving one point on 
the 1200 B. t. u. total 
heat curve. 

A few examples 
in the application of 
the chart suggest 
themselves. Assume 
steam to be formed 
at 103.38 lb. pres- 
sure ; required the 
necessary amount of 
superheat to be im- 
parted such that the 
steam shall be just 
dry after adiabatic 
expansion to atmos- 
pheric pressure. Let 
rs, Fig. 176, be the 
line of atmospheric pressure. Draw st vertically, intersecting di\ then 
t is the required initial condition. Along the adiabatic ts, the heat contents 
decrease from 1300 B. t. u. to 1150.4 B. t. u., a loss of 149.6 B. t. u. 

To find the condition of a mixture of unequal weights of water and super- 
heated steam after the establishment of thermal equilibrium, the whole 
operation being conducted at constant pressure : let the water, amounting 
to 10 lb., be at r, Fig. 176. Its heat contents are 1800 B. t. u. Let one 
pound of steam be at t, having the heat contents 1300 B. t. u. The heat 
gained by the water must equal that lost by the steam ; the final heat con- 
tents will then be 3100 B. t. u., or 282 B. t. u. per pound, and the state 




Fig. 17G. 



Arts. 398, 399, 401. — Entropy Diagram, Superheated 
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226 



APPLIED THERMODYNAMICS 



will be u, where the temperature is 312° F. ; the steam will have been 
completely liquefied. 

We may find, from the chart, the total heat in steam (wet, dry, or 
superheated) at any temperature, the quality and heat contents after 
adiabatic expansion from any initial to any final state, and the specific 
volume of saturated steam at any temperature and dryness. 

399. The Mollier Heat Chart. This is a variant on the temperature 
entropy diagram, in a form rather more convenient for some purposes. It 
has been developed by Thomas (41) to cover his experiments in the 
superheated region, as in Fig. 177. In this diagram, the vertical coordi- 
nate is entropy ; and the horizontal, total heat. The constant heat lines 
are thus vertical, while adiabatics are horizontal. The saturation curve 
is inclined upward to the right, and is concave toward the left. Lines of 
constant pressure are nearly continuous through the saturated and super- 
heated regions. The quality lines follow the curvature of the saturation 
line. The temperature lines in the superheated region are almost vertical. 
It should be remembered that the " total heat " thus used as a coordinate 
is nevertheless not a cardinal property. The " total heat " at t, Fig. 176, 
for example, is that quantity of heat which would have been imparted 
had water at 32° F. been converted into superheated steam at constant 
pressure. 

The total heat-pressure diagram (Fig. 185) is a diagram in which the coordi- 



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SATURATED STEAM TEMPERATURE. DEGREES F. 

Fig. 185. Art. 399, Problems. —Total Heat-pressure Diagram. 



480 



MATHEMATICAL THERMODYNAMICS 227 

nates are total heat above 32° F. and pressure; it usually includes curves of 
(a) constant volume, (b) constant dryness, (c) constant temperature. Vertical lines 
show the loss or gain of heat corresponding- to stated changes of volume or quality 
at constant pressure. Horizontal lines show the change in pressure, volume, and 
quality of steam resulting from throttling (Art. 387). This diagram is a use- 
ful supplement to that of Mollier. 

Vapors in General 

400. Analytical Method: Mathematical Thermodynamics. An expression 
for the volume of any saturated vapor was derived in Art. 368: 

W= V+ 11^- — - 
TdP 

Where the specific volume is known by experiment, this equation may be used for 
computing the latent heat. A general method of deriving this and certain related 
expressions is now to be described. Let a mixture of x lb. of dry vapor with 
(1 — x) lb. of liquid receive heat, dQ. Then 

dQ = JcxdT + c (1 - x) dT + Ldx, 

in which k is the "specific heat" of the continually dry vapor, L the latent heat 
of evaporation, and c the specific heat of the liquid. If P.V are the pressure and 
volume, and E the internal energy, in foot-pounds, of the mixture, then 

dQ = PdV + dE = ^dT + c (1 - x) dT + Ldx, whence 

778 

dE = 778 [kx + c (1 - x)~\ dT + 778 Ldx - PdV. 

Now V = (/) T, x; whence d V = f^ dT + ~ dx, whence 

oT ox 

dE = 778 [hx + c (1 - a;)] dT + 778 Ldx - P-*J- dT- P ^dx 

oT ox 

= J778 + c(l-z)] -Pf£} dT +(77SL-P^-)dx. 
{ oT j \ ox I 

Moreover, E = (/) T, x, whence 



o (oE\ S (8E\ ■ . 



U mL - pS £)=U 77SLkx+c{1 - i:)1 - p 



(all properties excepting V and x being functions of T only). 

The volume, V, may be written xu + v, where v is the volume of the liquid and 
u the increase of volume during vaporization. This gives &V = u8x or — = u. 

OX 



228 APPLIED THERMODYNAMICS 

s-2 y §2jr 
Also, since V= (/) T, x, = , and equation (A) becomes 

o Tox oxo T 

778 4^ - ^|^ = 778 (& - c), or 778 *£ + 778 (c - &) = u **, or 

dL . 7 u dP /rts 

\- c — k = (B) 

dT 778 dT K } 

Now if the heat is absorbed along any reversible path, ~ = dN, or 
,y_ kxdT + c(l - x)dT + Ldx _ kx + c(l - a:) , j, Z , 

-=(/)^w h ence > |(g)4 r (g). 

it -c dT 



T T* 

( 1L + c _ & = k (C) 

which may be combined with (B), giving 

77S L dT = = F _ . Art 369 , D) 

401. Computation of Properties. Equation (D), as thus derived, or as obtained 
in Art. 369, may be used to compute either the latent heat or the volume of any 
vapor when the other of these properties and the relation of temperature and pres- 
sure is known. The specific heat of the saturated vapor may be obtained from 
(C) ; the temperature of inversion is reached when the specific heat changes sign. 
For steam, if L = 1113.94 - 0.695 1 (Art. 379), where t is in degrees^F., or 

1113.94 - 0.695(7' - 459.6) where T is the absolute temperature: — = - 0.695. 

dl 

Also c = I; whence, from equation (C), k = 0.305 — — , which equals zero when 

r=1433° absolute* At 212° F., £ = 0.305 - —^ = -1.135. This may be roughly 

checked from Fig. 175. In Fig. 176, consider the path sb from 2.12° F. to 157° F., 

and from n — 1.735 ton = 1.835 (Fig. 175). The average height of the area csbe 

212 4- 157 
representing the heat absorbed is 459.6 H ^— = 644.1 ; whence, the area is 

644.1(1.835 — 1.735) = 64.41 B. t. u., and the mean specific heat between s and b is 
64.41 -^ (212 — 157) = 1.176. The properties of the volatile vapors used in refriger- 
ation are to some extent known only by computations of this sort. When once 
the pressure-temperature relation and the characteristic equation are ascertained by 
experiment, the other properties follow. 

* This would be the temperature of inversion of dry steam if the formula for L held : 
but L becomes zero at 689° F. (Art. 379), and the saturation curve for steam slopes downward 
toward the right throughout its entire extent. For the dry vapors of^chloroform and ben- 
zine, there exist known temperatures of inversion. 



VAPORS IN GENERAL 229 

402. Engineering Vapors. The properties of the vapors of steam, carbon 
dioxide, ammonia, sulphur dioxide, ether, alcohol, acetone, carbon disulphide, carbon 
tetrachloride, and chloroform have all been more or less thoroughly studied. The 
first five are of considerable importance. For ether, alcohol, chloroform, carbon disul- 
phide, carbon tetrachloride, and acetone, Zeuner has tabulated the pressure, tempera- 
ture, volume, total heat, latent heat, heat of the liquid, and internal and external 
work of vaporization, in both French and English units (42), on the basis of 
Regnault's experiments. The properties of these substances as given in Peabody's 
"Steam Tables" (1890) are reproduced from Zeuner, excepting that the values 
- 273.7 and 426.7 are used instead of - 273.0 and 424.0 for the location of the 
absolute zero centigrade and the centigrade mechanical equivalent of heat, 
respectively. Peabody's tables for these vapors are in French units only. Wood 
has derived expressions for the properties of these six vapors, but has not tabulated 
their values (43). Rankine (44) has tabulated the pressure, latent heat, and density 
of ether, per cubic foot, in English units, from Regnault's data. For carbon dioxide, 
the experimental results of Andrews, Cailletet and Hautefeuille, Cailletet and 
Mathias (45), and, finally, Amagat (46), have been collated by Mollier, whose 
table (47) of the properties of this vapor has been reproduced and extended, in 
French and English units, by Zeuner (48). The vapor tables appended to Chapter 
XVIII, it will be noted, are based on those of Zeuner. The entropy diagrams for am- 
monia, sulphur dioxide, and carbon dioxide, Figs. 314-316, have the same foundation. 

403. Ammonia. Anhydrous ammonia, largely used in refrigerating 
machines, was first studied by Regnault, who obtained the relation 

logp = 8.4079-—, 

in which p is in pounds per square foot and t is the absolute temperature. 
A " characteristic equation " between p, v, and t was derived by Ledoux 
(49) and employed by Zeuner to permit of the computation of V, L, e, r 
and the specific heat of the liquid (the last having recently been deter- 
mined experimentally (50)). The results thus derived were tabulated by 
Zeuner (51) for temperatures below 32° F. ; while for higher temperatures 
he uses the experimental values of Dietrici (52). Peabody's table (53), 
also derived from Ledoux, uses his values for temperatures exceeding 
32° F. ; Zeuner regards Ledoux's values in this region as unreliable. 
Peabody's table is in French units ; Zeuner's is in both. French and Eng- 
lish units. The latent heat of evaporation has been experimentally de- 
termined by Regnault (54) and Von Strombeck (55). The specific volume 
of the vapor at — 26.4° F. and atmospheric pressure is 17.51 cu. ft. ; that of 
the liquid is 0.025 ; whence from equation (D), Art. 400, 

L = ( V — v) — 

778 V J dT 



(17.51 - 0.025/ 2196 x 2 - 3026xl47xl44 V555 ? 

V - J \ 422 9 v 422 9 / 



433.2 
778 v ^ ~~'V 433.2x433.2 



230 APPLIED THERMODYNAMICS 

dP 

the value of — being obtained by differentiating Regnault's equation, 

above given. From a study of Regnault's experiments, Wood has derived 
the characteristic equation, 

PF = 91 _16920 

T jiyosi ' 

which is the basis of his table of the properties of ammonia vapor (56). 
Wood's table agrees quite closely with Zeuner's, as to the relation between 
pressure and temperature ; but his value of L is much less variable. For 
temperatures below 0° C, the specific volumes given by Wood are rather 
less than those by Zeuner; for higher temperatures, the volumes vary 
less. Zeuner's table must be regarded as probably more reliable. The 
specific heat (0.508) and the density (0.597, when air = 1.0) of the super- 
heated vapor have been determined by experiment. 

404. Sulphur Dioxide. The specific heat of the superheated vapor is given by 
Regnault as 0.15438 (57). The specific volume, as compared with that of air, is 
2.23 (58). The specific volume of the liquid is 0.0007 (59) ; its specific heat is 
approximately 0.4. A characteristic equation for the saturated vapor has been 
derived from Regnault's experiments : 

PV= 26.4 T- 184 P - 22 ; 

in which P is in pounds per square foot, Fin cubic feet per pound, and T in abso- 
lute degrees. The relation between pressure and temperature has been studied by 
Regnault, Sajotschewski, Blumcke, and Miller. Regnault's observations were 
made between — 40° and 149° F. ; Miller's, between 68 and 211° F. ; a table repre- 
senting the combined results has been given by Miller (60). In the usual form 
of the general equation, 

log p = a — bd n — ce n , 

the values given by Peabody for pressures in pounds per square inch are (61) 
a = 3.9527847, log b = 0.4792425, log d = 1.9984994, log c = 1.1659562, log e = 
1.99293890, n — 18.4 + Fahrenheit temperature. The specific volumes, determined 
by the characteristic equation and the pressure-temperature formula, permit of the 
computation of the latent heat from equation (D), Art. 400. An empirical formula 
for this property is L = 176 — 0.27 (t — 32), in which t is the Fahrenheit tempera- 
ture. The experimental results of Cailletet and Mathias, and of Mathias alone (62), 
have led to the tables of Zeuner (63). Peabody, following Ledoux's analysis, has 
also tabulated the properties in French units. Wood (64) has independently com- 
puted the properties in both French and English units. Comparing Wood's, Zeu- 
ner's, and Peabody's tables, Zeuner's values for L and V are both less than those of 
Peabody. At 0° F., he makes L less than does Wood, departing even more widely 
than the latter from Jacobus' experimental results (65) ; at 30° F., his value of L is 
greater than Wood's, and at 104° F., it is again less. The tabulated values of the 
specific volumes differ correspondingly. Zeuner's table may be regarded as sus- 



STEAM PLANT CYCLE 



231 



tained by the experiments of Cailletet and Mathias, but the lack of concordance 
with the experimental results of Jacobus remains to be explained. 

405. Steam at Low Temperatures. Ordinary tables do not give the proper- 
ties of water vapor for temperatures lower than those corresponding to the abso- 
lute pressures reached in steam engineering. Zeuner has, however, tabulated 
them for temperatures down to — 4° F. (66). 

Steam Cycles 

406. The Carnot Cycle for Steam. This is shown in Figs. 163, 
179. The efficiency of the cycle abed may be read from the entropy 

T-t 



diagram as 



T 



The external 



work done per pound of steam 

T — t 

is L — - — ; or if the steam at b 

T-t 



is wet, it is xL 



If the 



T 
















a 


/ 




, V 




ml 


1 


a 




• 

i 
i 

i 
i 

c 








i 

i 





fluid at the beginning of the 
cycle (point a) is wet steam 
instead of water, the dryness 
being x M then the work per 

_ u , . Fig. 179. Art. 406. —Carnot Cycle for Steam. 

pound oi steam is L(x — x ) 

T — t 

— — — • In the cycle first discussed, in order that the final adiabatic 

compression may bring the substance back to its initially dry state at 
a, such compression must begin at d, where the dryness is md -f- mn. 
p The Carnot cycle is impracticable 

with steam ; the substance at d is 
mostly liquid, and cannot be raised 
in temperature by compression. 
What is actually done is to allow 
condensation along cd to be com- 
pleted, and then to Avarm the liquid 
or its equivalent along ma by trans- 
mission of heat from an external 
source. This, of course, lowers 
the efficiency. 

407. The Steam Power Plant. The cycle is then not completed in 
the cylinder of the engine. In Fig. 180, let the substance at d be 





BOILER 




Nj 


d 


CONDENSER 







Fig. 180. Arts. 407, 408, 410, 412, 413 
The Steam Power Plant. 



232 



APPLIED THERMODYNAMICS 



oold water, either that resulting from the action of the condenser 
on the fluid which has passed through the engine, or an external 
supply. This water is now delivered by the feed pump to the boiler, 
in which its temperature and pressure become those along ah. The 
work done by the feed pump per pound of fluid is that of raising 
unit weight of the liquid against a head equivalent to the pressure ; 
or, what is the same thing, the product of the specific volume of the 
water by the range in pressure, in pounds per square foot. From 
a to b the substance is in the boiler, being changed 'from water to 
steam. Along be, it is expanding in the cylinder; along cd it is 
being liquefied in the condenser or being discharged to the atmos- 
phere. In the former case, the resulting liquid reaches the feed 
pump at d. In the latter, a fresh supply of liquid is taken in at d, 
but this may be thermally equivalent to the liquid resulting from 
atmospheric exhaust along cd. (See footnote, Art. 502.) The four 

organs, feed pump, boiler, cylinder, 
and condenser, are those essential in 
a steam power plant. The cycle rep- 
resents the changes undergone by 
the fluid in its passage through them. 



T 

he Tc 


l& 


I ; i > 


\u 


1 / / y* 


9 \f 


71 « 


,-y 


i j m ^ 


< 



408. Clausius Cycle. The cycle 
of Fig. 180, worked without adiabatic 
compression, is known as that of 
Clausius. Its entropy diagram is 
shown as debc in Fig. 181, that of 
the corresponding Carnot cycle being 
dhbe. The Carnot efficiency is obviously greater than that of the 
Clausius cycle. For wet steam the corresponding cycles are dekl 
and dhkl. 



Fig. 181. 



Arts. 408-413. — Steam 
Cycles. 



409. Efficiency. In Fig. 181, cycle debc, the efficiency is 

debc idej +jebK— idcK _ h e — h d +L b — x c L f 
idebK idej +jebK h e — h d + L b 

J df L f +T d 



, if the specific heat of the 



RANKINE CYCLE 233 

liquid be unity. Then letting T, L refer to the state 6, and £, I to 
the state c, the efficiency is 

T-t+L = T-t + L ' 

which is determined solely by the temperature limits T and t. For 
steam initially wet, the efficiency is 

T-t + XL 

410. Work Area. In Figs. 180, 181, we have 
W= W ab +W bc - W cd -W da 

= [pb(?b -*«)] + (A + n - K - x c ri) - [p c Zc(v f - v d y\ - o, 

ignoring the small amount of work done by the feed pump in forcing 

the liquid into the boiler. But p b (y h — v a ) = e b and p c x c (y f — v d ") = x c e f 

(Art. 359), whence 

W= h e + L b -h d - x c L f , 

a result identical with the numerator of the first expression in Art. 
409. 

411. Rankine Cycle. The cycle debgq, Fig. 181, abgqd, Fig. 180, 
is known as that of Rankine (67). It differs from that of Clausius 
merely in that expansion is incomplete, the " toe " gcq, Fig. 180, 
being cut off by the limiting cylinder volume line gq. This is the 
ideal cycle nearest which actual steam engines work. The line gq in 
Fig. 181 is plotted as a line of constant volume (Art. 377). The 
efficiency is obviously less than that of the Clausius cycle ; it is 

ebgqd _ W ab + W H - W qi (Fig. 180) 
idebK h e — h d + L b 

= [Pb(vo ~ *Q] + Qi e + r b -h r - x g r s ) - [p q x q (v f - v d )] m 
h e -h d + L b 

The values of A r , x g , r„ x g , depend upon the limiting volume v g = v q , 
and may be most readily ascertained by inspecting Fig. 175. The 
computation of these properties resolves itself into the problem : given 



234 APPLIED THERMODYNAMICS 

the initial state, to find the temperature after adiabatic expansion to a 
given volume. We have 

v g - v r = x g (y s - iv), n g = n» 

\ g e ^1 + ^ 
x _ %-n r = n b -n r _ e T r T e 

9 n s — n r n s - n r L s + T r '• 
whence 

, r < l0 4 + f) r , 

v g = v r + j (v, - O, 

in which v g , T e , L b are given, v r = 0.017, and v s , L s are functions of 
2L the value of which is to be ascertained. 



412. Non-expansive Cycle. This appears as debt, Fig. 181 ; and abed, Fig. 180. 
No expansion occurs ; work is done only as steam is evaporated or condensed. 
The efficiency is (Fig. 181) 

debt = W ab - W ed (Fig. 180) = p b (v b - v a ) - p t (v b - v d ) 
idebK h e - h d + L b h e - li d + L b 

This is the least efficient of the cycles considered. 



413. Pambour Cycle. The cycle debf, Fig. 181, represents the operation of a 
plant in which the steam remains dry throughout expansion. It is called the 
Pambour cycle. Expansion may be incomplete, giving such a diagram as debuq. 
Let abed in Fig. 180 represent debf in Fig. 181. The efficiency is 

external work done external work done 



gross heat absorbed heat rejected -f external work done 

_ W ab + W bc - W cd _ p b (v b - v a ) + 16( p b v b -p f v f ) - p f (v f - v d ) 
L f + Wab + W bc - W cd L f + p b (v b - v a ) + l§(p b v b -P/V/) - M y /~ v dY 

in which the saturation curve bf may be represented by the formula joyie — con- 
stant (Art. 363). A second method for computing the efficiency is as follows: 

the area debf = I — dT, in which T and t are the temperatures along eb and df 
respectively, and L=(f)T = 1433 - 0.695 T (Art. 379). This gives 

debf= 1433 log e - - 0.695 (T - t), 

and the efficiency is 

14331og e f-0.695(r-0 
debf debf t 



idebfv debf + idfv H33 ]oge T _ Q 6Q5( r _ t) + L/ 



SUPERHEATED CYCLES 



235 



The two computations will not precisely agree, because the exponent \% does not 
exactly represent the saturation curve, nor does the formula for L in terms of T 
hold rigorously. 

Of the whole amount of heat supplied, the portion Kbfv was added 
during expansion, as by a steam jacket (Art. 439). To ascertain this 
amount, we have 

heat added by jacket 

= whole heat supplied — heat present at beginning of expansion- 

= 1433 log e - - 0.695 ( T - 1) + L f - h e + h d - L b . 

The efficiency is apparently v less than that of the Olausius cycle (Fig. 
181). In practice, however, steam jacketing increases the efficiency of 
engines, for reasons which will appear (Art. 439). 




414. Cycles with Superheat. As in Art. 397, three cases are pos- 
sible. Figure 182 shows the Clausius cycles debxw, debyf, debzAf, 
in which the steam is respectively wet, dry, and superheated at. the 
end of expansion. To appreciate 
the gain in efficiency due to super- 
heat, compare the first of these 
cycles, not with the dry steam 
Clausius cycle debt; but with the 
superior Oarnot cycle dhbc. If the 
path of superheating were bC, the 
efficiency would be unchanged ; 
the actual path is bx, and the work 
area bx C is gained at 100 per cent 
efficiency. The cycle debxw is 
thus more efficient than the Car- 
not cycle dhbc, and consequently 
still more efficient than the Clausius cycle debc. It is not more 
efficient than a Cariiot cycle through its own temperature limits. 

The cycle debyf shows a further gain in efficiency, the work area 
added at 100 per cent effectiveness being byE. The cycle debzAf 
shows a still greater addition of this desirable work area, but a loss of 
area AfB now appears. Maximum efficiency appears to be secured 
with such a cycle as the second of those considered, in which the 
steam is about dry at the end of expansion. The Carnot formula 




Fig 



182. Art. 414. — Cycles with 
Superheat. 



236 APPLIED THERMODYNAMICS 

suggests the desirability of a high upper temperature, and superheating 
leads to this ; but when superheating is carried so far as to appreciably 
raise the temperature of heat emission, as in the cycle debzAf, the 
efficiency begins to fall. 

415. Efficiencies. The work areas of the three cycles discussed 
may be thus expressed : 

"debxw — J^debxw = **de + ■"«& + ^bx ~ -**wd 

= h e -h d + L b + h x (T x - T b )-x w L f \ 

"debyf = H-debyf = ^de + -"eft + -"fty ~~ -"/a 

= h e -h d + L b + k 2 (T y -T b )-L / ; 

" dehzAf = H debzAf = H de -f- H eb -f R bz — H Af — Hf d 

= K-h + L b + k B <:T z -T 6 )-}c i (T A -T f )-Z f ; 

in which Jc v k v Jc 3 , k±, refer to the mean specific heats over the re- 
spective pressure and temperature ranges. The efficiencies are 
obtained by dividing these expressions by the gross amounts of heat 
absorbed. The equations given in Art. 397 permit of computation 
of such quantities as are not assumed. 

416. Itemized External Work. The pressure and temperature at the 
beginning of expansion being given, the volume may be computed and 
the external work during the reception of heat expressed in terms of 
P and V, The temperature or pressure at the end of expansion being 
given, the volume may be computed and the negative external work 
during the rejection of heat calculated in similar terms. The whole 
work of the cycle, less the algebraic sum of these two work quantities 
(the feed pump work being ignored), equals the work under the 
adiabatic, which may be approximately checked from the formula 

ZLR- , a suitable value being used for n (Art. 394). A second 

n — 1 

approximation may be made by taking the adiabatic work as equivalent 

to the decrease in internal' energy, which at any superheated state has 

k 
the value h + r 4- -(T— f), T being the actual temperature, and h, r, 

y 

t referring to the condition of saturated steam at the stated pressure. 
The most simple method of obtaining the total work of the cycle is to 



COMPARISONS 



237 



read from Fig. 177 the " total heat" values at the beginning and end of 
expansion. 

417. Comparison of Cycles. In Fig. 183, we have the following 
cycles : 

T 



h 






k 


\ 


f -H 


5 

M 


T 


r 
27 / 




1^ 

\a 


L 






/ 


' /, 


^ 


■^>^9 






*/ 


/ 


/^? 






V 










/ A 




J 








d 




K/ ///// 




I 






\ 




b>^ 




tpswq 








\ 


/ 


I 






2 




3 



Fig. 183. Arts. 417, 441, 442. — Seventeen Steam Cycles. 



Clausius, 



Rankine, 

Non-expansive, 

Pambour, 



with dry steam, debc (the corresponding Carnot 

cycle being dhbe); 
with wet steam, dekl; 
with dry steam, debgq ; 
with wet steam, dekJq; 
with dry steam, debt ; 
with wet steam, deJcIC; 
complete expansion, debf; 
incomplete expansion, debug; 
Superheated to x, complete expansion, debxw ; 

incomplete expansion, debxLuq; 
no expansion, debxNp; 
Superheated to y, complete expansion, debyf\ 

incomplete expansion, debyMuq; 
no expansion, debi/Rs ; 
Superheated to g, complete expansion, debzAf; 
incomplete expansion, debzTuq; 
no expansion, debzVw. 



238 



APPLIED THERMODYNAMICS 



The lines ib, pNx> sRy, wVz, quT, are lines of constant volume. 
Superheating without expansion would be unwise on either technical 
or practical grounds ; superheating with incomplete expansion is the 
condition of universal practice in reciprocating engines. The 
seventeen cycles are drawn to P V coordinates in Fig. 184. 



e 


k b x y z 












n\ 










- 






^ 









T 

l\\ 




ci 




t 


P 


s 




w 




J ^^$^ 


^^^"^A 



Fig. 184. Arts. 417, 423, 424, 517. — Seventeen Steam Cycles. 



Illustrative Problem 

To compare the efficiencies, and the cyclic areas as related to the maximum volume at- 
tained: let the maximum pressure be 140 lb., the minimum pressure 2 lb., and consider 
the Clausius cycle (a) with steam initially dry, (b) with steam initially 90 per cent 
dry; the Rankine with initially dry steam and a maximum volume of 13 cu. ft., 
the same Rankine with steam initially 90 per cent dry ; the non-expansive 
with steam dry and 90 per cent dry ; the Pambour (a) with complete expansion 
and (b) with a maximum volume of 13 cu. ft. ; and the nine types of superheated 
cycle, the steam being ; (a) 96 per cent dry, (b) dry, (c) 40° F. superheated, at the 
end of complete expansion ; and expansion being (a) complete, (b) limited to a 
maximum volume of 13 cu. ft., (c) eliminated. 

I. Clausius cycle. The gross heat absorbed is h UQ — h 2 + L H0 =324:.6 — 94.0 + 867.6 

= 1098.2. 
The dryness at the end of expansion is dc -=- df, Fig. 183, = (n e — n d + n eb ) ~ n df 

= (0.5072 - 0.1749 + 1.0675) - 1.7431 = 0.803. 
The heat rejected along cd is x c L f = 0.803 x 1021 = 819. 4. 



The work done is 1098.2 - 819.4 



,8B. t. u. The efficiency is 



278.8 
1098.2 



= 0.254. 



The efficiency of the corresponding Carnot cycle is 
353.1-126.15 



T —T 
T 



353.1 + 459.( 



0. 



II. Clausius cycle with wet steam. The gross heat absorbed is h 14() — h 2 + x k L U0 
= 324.6 - 94.0 + (0.90 x 867.6) = 1015.44- 
The dryness at the end of expansion is dl -=- df= (n e — n d + n ek ) -r- n d} 
= (0.5072 - 0.1749 + 0.90 x 1.0675) -j- 1.7431 = OJ41. 



COMPARISONS 239 

The heat rejected along Id is x l L f = 0.741 x 1021 = 756. 

The work done is 1015.44 - 756 = 259. U B. t. u. 

259 44 
The efficiency is m5U = 0j5 4- 

(It is in all cases somewhat less than that of the initially dry steam cycle.) 

III. Rankine cycle, dry steam. The gross heat absorbed, as in I, is 1098.2. 

The workalong de, Fig. 184,is 144 x 138 x 0.017 = 338.5 foot-pounds (Art. 407); 
along eb is 144 x 140 x (V b — 0.017) = 64,300 foot-pounds ; 

(V b = 3.219) 
along bg is h e + r h — h z — x g r g = 109.76 B. t. u. 

(From Fig. 175, ^ = 247° F., whence L g = 94:7 A, F a = 14.52, x g = }f'?^^}L 

= 0.895.) 

n Zg _ n e — n z Ar n eb 

n Za l^a -f- 1 a 

= [0.5072-2.3 (log T q - log 491.6) + 1.0675] T q 
1433 - 0.695 T g 
For T g = 247° F. = 706.6° absolute, this equation gives x g = 0.905 ; a suffi- 
cient check, considering that Fig. 175 is based on a different set of values 
than those used in the steam table. Then h z = 215.4, r g = 871.6. 
The work along qd is P d ( V q - V d ) = 144 x 2 x (13 - 0.017) = 37 40 foot- 
pounds. 
The whole work of the cycle is 64300 - 338.5 - 3740 109>76 = m Jg B ^ 
J U 778 

The efficiency is 187/29 = 0.1704- 
d J 1098.2 

IV. Rankine cycle, wet steam. The gross heat absorbed is as in II, 1015.44- 

The negative work along de and qd is, as in III, 338.5 + 3740 — 4078.5 foot- 
pounds. 
The work along ek is 144 x 140 x 0.90( V b - 0.017)= 57,870 foot-pounds. 
The work along kJ is h e + x k r b — h x — Xjr T = 99.8 B. t. u. 
(From Fig. 175, t x = 242° F., whence h x = 210.3, r Y = 875.3, V T = 15.78, 

13- 0-617, = 0,826.) 

15.78-0.017 / 

The whole work of the cycle is ° 7870 ~ 4078 j + 99.8 = 169.1 B. t. u. 

J J 778 

The efficiency is 169J = 0.1667. 
M J 1015.44 

V. Non-expansive cycle, dry steam. The gross heat absorbed, as in I, is 1098.2. 
The work along de, as in III, is 338.5 foot-pounds ; 
along eb, as in III, is 64,300 foot-pounds ; 

along td is p d (V h - V d ) = 144 x 2 x (3.219 - 0.017)= 922 foot-pounds. 
The wAo£e ?yor& of the cycle is 

64,300 - 338.5 - 922 = 63,039.5 foot-pounds = 81.05 B. t. u. 

The efficiency is 810 ° = 0.074. 
M y 1098.2 



240 APPLIED THERMODYNAMICS 

VI. Non-expansive cycle, wet steam. The gross heal absorbed, as in II, is 10 15 44. 
The work along de, ek, as in IV, is - 338.5 + 57,870 = 57,531.5 foot-pounds. 
> The work along Kd is 

Pd(V K - 0.017)= 144 x 2 x 0.90 x (3.219 - 0.17)= 82 9. 8 foot-pounds. 
The whole work of the cycle is 

57,531.5 - 829.8 = 56,701.7 foot-pounds = 73 B. t. u. 

The efficiency is = 0.0722. 

M * 1015.44 

VII. Pambour cycle, complete expansion. The heat rejected is L f — 1021.0. 

The work along de, eb, as in III, is - 338.5 + 64300 = 63,961.5 foot-pounds. 
The work along bf is 

P>V :Zx /V ' = Ui { (U0 X 8J8 ll -"I" X 173 ' 5) ) = ^^foot-pounds. 

The work along fd is P d ( V f - Va) = 2 x 144 (173.5 - 0.017) = 49,900 foot- 
pounds. 

The whole work of the cycle is 63,961.5 + 236,800 - 49,900 = 250,861.5 foot- 
pounds. 

(Otherwise 1433 log,— - 0.695 (T - t) = 312 B. t. u. = 242,000 foot-pounds 

(Art. 413).) l 

Using a mean of the two values for the whole work, the gross heat absorbed 

is ^^ + 1021 = 1340 B. t. u. and the efficiency is 246430 = 0.238. 
778 M * 778 x 1340 

The heat supplied by the jacket is 1340 - 1098.2 = 246.8 B. t. u. 

VIII. Pambour cycle, incomplete expansion (debuq). In this case, we cannot 
directly find the heat rejected, nor can we obtain the work area by inte- 
gration* From Fig. 175 (or from the steam table), we find T u = 253.8° F., 
P u = 31.84. The heat area under bu is then, very nearly, 

r " + Tb (n u - 0= 712 - 6 + 812 - 7 (1.6953 - 1.5747) = 92 B. t. u. 

£ — 

The whole heat absorbed is then 1098.2 + 92 = 1190.2 B. t. u. 

The work along de, eb, as in VII, is 63,961.5 foot-pounds. 

The work along bu is 144 x 16[(140 x 3.219)- (31.84 x 13)] = 85,800 foot- 
pounds. 

The work along qd, as in III, is 3740 foot-pounds. 

The whole work of the cycle is 

63,961.5 + 85,800 - 3740 = 146,021.5 foot-pounds = 188.2 B. t. u. 

1009 
The efficiency is = 0.1585. 

M y 1190.2 

* A satisfactory solution may be had by obtaining the area of the cycle in two parts, a 
horizontal line being drawn through u to de. The upper part may then be treated as a com- 
plete-expansion Pambour cycle and the lower as a non-expansive cycle. The gross heat 
absorbed is equal to the work of the upper cycle plus the latent heat of vaporization at the 
division temperature plus the difference of the heats of liquid at the division temperature 
and the lowest temperature. 

A somewhat similar treatment leads to a general solution for any Rankine cycle : in 
which, if the temperature at the end of expansion be given, the use of charts becomes 
unnecessary. 



COMPARISONS 241 

IX. Superheated cycle, steam 0.96 dry at the end of expansion ; complete expansion; 
cycle debxw. We have ?i w = n d +x w n d/ = 0.1749 + (0.96 x 1.7431) = 1.8449. 
The state x(n x = n w ) may now be found either from Fig. 175 or from the 
superheated steam table. Using the last, we find jT x = 931.1° F., H^ 1481.8, 
V x = 5.96. The whole heat absorbed, measured above T d , is then 
1481.8 - 94.0 = 1387.8. 
The heat rejected is x w L f = 0.96 x 1021 = 981. 
The external work done is 1387.8 — 981 = 406.8, and the efficiency is 

1387.8 
(The efficiency of the Carnot cycle within the same temperature limits is 
931.1- 126.15 
931.1 + 459.6 J 

X. The same superheated cycle, with incomplete expansion. 
The whole heat absorbed, as before, is 1387.8. 
The work done along de, eb, as in III, is 63,961.5 foot-pounds. 
The ivork done along bx is 

P h (V x - Vi)= 144 x 140(5.96 - 3.219)= 55,000 foot-pounds. 
The icork done along xL is 

P ' V ' n ~_* LVL = 144 (Q40x5.98)- 8 (51.1xl3)^ = 81 j 00 foot _ pounds . 

(V L = 13, P X V X ^™ = P L V L im , P L = Uo(^®Y m = 51.1 ; a procedure 

which is, however, only approximately correct (Art. 391).) 
The work along qd, as in III, is 3740 foot-pounds. 
The whole work of the cycle is 
63,961.5 +55,000 + 81,500 - 3740 = 196,721.5 foot-pounds = 253.5 B. t. u. 

2*53 5 

The efficiency is " ' ; = 0.183. 
lob/. 8 

XI. The same superheated cycle, icorked non-expansively. The gross heat absorbed 
is 1387.8. 
The work along de, eb, bx, as in X, is 118,961.5 foot-pounds. 
The work along pd is 2 x 144 x (5.96 - 0.017) = 1716 foot-pounds. 
The whole work of the cycle is 

118,961.5 - 1716 = 117,245.5 foot-pounds = 150.6 B. t. u. 

The efficiency is 150 ' 6 = 0.1086. 
1387.8 

XII. Superheated cycle, steam dry at the end of expansion, complete expansion ; cycle 
debyf. 
We have n y = n f — 1.918. This makes the temperature at y above the 
range of our table. Figure 171 shows, however, that at high tempera- 
tures the variations in the mean value of k are less marked. We may 
perhaps then extrapolate values in the superheated steam table, giving 
T y = 1120.1° F., H y = 1573.5, V y = 6.81. The whole heat absorbed, above 
T d , is then 1573.5 - 94.0 = 1479.5. The heat rejected is L f = 1021. 



242 APPLIED THERMODYNAMICS 

The external work done is 1479.5 — 1021 = 458.5 B. t. u., and the efficiency 

i. «§£ = «* 

1479.5 

XIII. Superheated cycle as above, but with incomplete expansion. The gross heat 

absorbed is 1479.5. 
The work done along de, eb, as in III, is 63,961.5 foot-pounds. 
The work done along by is 144 x 140 x (6.81 - 3.219) = 72,200 foot-pounds. 

/Q g1 \ 1.298 

The pressure at M is 140 f - 1 — - J = 60.3 pounds, approximately. 

The work done along yM is W ( 14 ° * 6 - 81 )-( 60 - 3 * 13 ) \ _ 81yl00 f oot . 

pounds, also approximately. 
The work done along qd, as in III, is 3740 foot-pounds. 
The whole work of the cycle is 

63,961.5 + 72,200 + 81,100 - 3740 = 213,521.5 foot-pounds = 275 B. t. u. 

97^ 

The efficiency is f' = 0.187. 
M u 1479.5 

XIV. Superheated cycle as above, but without expansion. The gross heat absorbed 

is 1479.5. 
The work along de, eb, by, as in XIII, is 136,161.5 foot-pounds. 
The work along sd is 2 x 144 x (6.81 - 0.017) = 1952 foot-pounds. 
The total work is 136,161.5 - 1952 = 134,209.5 foot-pounds = 172.7 B. t. u. 

I79 7 
The efficiency is — -f 1 — = 0.117. 
M J 1479.5 

XV. Superheated cycle, steam superheated 40° F. at the end of expansion; expan- 
sion complete: cycle debzAf. We have n A — n z — 1.9186. A rather 
doubtful extrapolation now makes T z = 1202.1° F., H z = 1613.4, Y 2 
= 7.18. The whole heat absorbed is 1613.4 - 91.0 = 1519.4. The heat re- 
jected is H A = 1133.2. The total work is 1519.4 - 1133.2 = 386.2 B. t. u., 

QQf* Q 

and the efficiency is — — ^ = 0.255. 
M u 1519.1 

XVI. The same superheated cycle, ivith incomplete expansion. The pressure at T is 

/ 7 18 \ 1 - 298 
lliOf- 1 ^—) —65.3 pounds. The work along zT (approximately) is 

144 / (140 x 7.18)-(65.3 x 13) \ _ 72 ^900 foot-pounds. The whole work is 

63,961.5 + [144 x 140 x (7.18 - 3.219)] + 73,900 - 3740 = 213,921.5 foot- 

■ 975 3 

pounds = 275.3 B. t. u., and the efficiency is -^- — '— = 0.182. 

XVII. The same superheated cycle without expansion. The total work is 63,961.5 + 
[144 x 140 x (7.18 - 3.219)] - [2 x 144 x (7.18 - 0.017)] = 141,701.5 foot- 
pounds = 182.2 B. t. u., and the efficiency is 0.1203. 

418. Discussion of Results. The saturated steam cycles rank in 
order of efficiency as follows: Carnot, 0.28; Clausius, with dry steam, 



COMPARISONS 



243 



0.254 ; with wet steam, 0.254 (a greater percentage of initial wetness 
would have perceptibly reduced the efficiency) ; Pambour, with com- 
plete expansion, 0.238 ; with incomplete expansion, 0.1585 ; Rankine, 
with dry steam, 0.1704 ; with wet steam, 0.1667; non-expansive, with 
dry steam 0.074; with wet steam, 0.0722. The economical impor- 
tance of using initially dry steam and as much expansion as possible 
is evident. The Pambour type of cycle has nothing to commend it, 
the average temperature at which heat is received being lowered. 
The Rankine cycle is necessarily one of low efficiency at low expan- 
sion, the non-expansive cycle showing the maximum waste. 

Comparing the superheated cycles, we have the following 
efficiencies : 



Cycle 


Complete Expansion 


Incomplete Expansion 


No Expansion 


debxw 
debyf 
debzAf 


0.293 

0.31 

0.255 


0.183 
0.187 
0.182 


0.1086 

0.117 

0.1203 



The approximations used in solution* will not invalidate the 
conclusions (a) that superheating gives highest efficiency when it is 
carried to such an extent that the steam is about dry at the end of 
complete expansion; (7>) that incomplete expansion seriously re- 
duces the efficiency ; (<?) that in a non-expansive cycle the effi- 
ciency increases indefinitely with the amount of superheating. As 
a general conclusion, the economical development of the steam en- 
gine seems to be most easily possible by the use of a superheated 
cycle of the finally-dry-steam type, with as much expansion as pos- 
sible. We shall discuss in Chapter XIII what practical modifica- 
tions, if any, must be applied to this conclusion. 

The limiting volumes of the various cycles are 
V c for the Carnot, I, = 139.3. 
V % for 11 = 128.2. 

V u = V g for III, IV, VIII, X, XIII, XVI = 13. 0. 
V b for V = 3.219. 
V k for VI = 2.9. 
V f for VII, XII = 173.5. 

* See footnote, Problem 53, page 255 



v w 


for IX 


= 166.5. 


v x 


for XI 


= 5.96. 


v v 


for XIV 


= 6.81. 


v A 


for XV 


= 186.1. 


V z for XVII 


= 7.18. 



244 



APPLIED THERMODYNAMICS 



The capacity of an engine of given dimensions is proportional to 



cyclic area 



maximum volume 



, which quotient has the following values' 



Carnot, temperature range x entropy range 



= 226.95(1.5747 - 0.1749)= 317.5: quotient = 



317.5 



I. 

II. 

III. 

IV. 

V. 

VI. 

VII. 

VIII. 

IX. 



278.8-; 

259.44 

187.29 

169.1- 

81.05-^ 



139.3 = 2.00. 
f- 128.2 = 2.015. 
- 13 = 14.4. 
13=13.0. 
3.219 = 25.1. 



73.0-2.9=25.1. 
318 -v- 173.5 = 1.84. 

188.2-13 = 14.5. 
406.8-*- 166.5 = 2.445. 



139.3 

X. 253.5 -v- 13 = 19.45. 

XI. 150.6-5.96 = 25.3. 

XII. 458.5 -j- 173.5 = 2.65. 

XIII. 275^-13 = 21.1. 

XIV. 172.7-6.81=25.4. 
XV. 386.2-186.1 = 2.075 

XVI. 275.3-13 = 21.1. 

XVII. 182.2-7.18 = 25.5. 



2.29. 



Here we find a variation much greater than is the case with the 
efficiencies ; but the values may be considered in three groups, the 
first including the five non-expansive cycles, giving maximum 
capacity (and minimum efficiency) ; the second including the six 
cycles with incomplete expansion, in which the capacity varies from 
13 to 21.1 and the efficiency from 0.1585 to 0.187; and the third 
including six cycles of maximum efficiency but of minimum capacity, 
ranging from 1.84 to 2.65. In this group, fortunately, the cycle of 
maximum efficiency (XII) is also that of maximum capacity. 



* The assumption of a constant limiting volume line Tuq, Fig. 183, is scarcely 
fair to the superheated steam cycles. In practice, either the ratio of expansion or the 
amount of constant volume pressure-drop at the end of expansion is assumed. As the 
first increases and the second decreases, the economy increases and the capacity figure 
decreases. The following table suggests that with either an equal pressure drop or an 
equal expansion ratio the efficiencies of the superheated cycles would compare still 
more favorably with that of the Rankine : — 

Cycles with Incomplete Expansion 



Cycle 


Eatio of Expansion 


Pressure Drop 


Rankine 
Superheat I 
Superheat II 
Superheat III 


V g - V b = 13 - 3.219 = 4.04 
V L - V x = 13 - 5.9G = 2.185 
V M +V y = 13 -6.81 =1.91 
V T + F*=13-7.18 =1.815 


P g - p q = 26.3 
P L -P q = 49.1 
P M -P q = 58.3 
P T -P g = 63.3 



THE STEAM TABLES 245 

Practically, high efficiency means fuel saving and high capacity 
means economy in the first cost of the engine. The general incom- 
patibility of the two affords a fundamental commercial problem in 
steam engine design, it being the function of the engineer to estab- 
lish a compromise. 

419. The Ideal Steam Engine. No engine nsing saturated steam can develop 
an efficiency greater than that of the Clansius cycle, the attainable temperature 
limits in present practice being between 100° and 400° F., or, for non-condensing 
engines, between 212° F. and 400° F. The steam engine is inherently a wasteful 
machine ; the wastes of practice, not thus far considered in dealing with the ideal 
cycle, are treated with in the succeeding chapter. 

The Steam Tables 

420. Saturated Steam. The table on pages 247, 248 is abridged from Marks' 
and Davis' Tables and Diagrams (18). In computing these, the absolute zero 
was taken at — 459.64° F. ; the values of h and n w were obtained from the experi- 
ments of Barnes and Dietrici (68) on the specific heat of water; the mechanical 
equivalent of heat was taken at 777.52 ; the pressure-temperature relation as found 
by Holborn and Henning (Art. 360) ; the thermal unit is the "mean B. t. u." (see- 
footnote, Art. 23) ; the value of H is as in Art. 388 ; and the specific volumes 
were computed as in Art. 368. The symbols have the following significance: — 

P = pressure in pounds per square inch, absolute ; 
T = temperature Fahrenheit ; 
V — volume of one pound, cubic feet ; 
h = heat in the liquid above 32° F., B. t. u. ; 
H = total heat above 32° F., B. t. u. ; 
L = heat of vaporization = H — h, B. t. u. ; 

r = disgregation work of vaporization = L — e (Art. 359), B. t. u. ; 
n w = entropy of the liquid at the boiling point, above 32° F. ; 

n e = entropy of vaporization = — ; 

n s = total entropy of the dry vapor = n -f n e . 

421. Superheated Steam. The computations of Art. 417 may suggest the 
amount of labor involved in solving problems involving superheated steam. This 
is largely due to the fact that the specific heat of superheated steam is variable. 
Figure 177, representing Thomas' experiments, may be employed for calculations 
which do not include volumes; and volumes may be in some cases dealt with by 
the Linde formula (Art. 363). The most convenient procedure is to use a table, 
such as that of Heck (71), or of Marks and Davis, in the work already referred to. 
On the following page is an extract from the latter table. The values of k used 
are the result of a harmonization of the determinations of Knoblauch and Jakob 
(Art. 384) and Holborn and Henning (69) and other data (70). They differ 
somewhat from those given in Fig. 170. The total heat values are obtained by 



246 



APPLIED THERMODYNAMICS 



adding the values of k(*T — t) over successive short intervals of temperature to 
the total heat at saturation ; the entropy is computed in a corresponding manner. 
The specific volumes are from the Linde formula. 



PROPERTIES OF SUPERHEATED STEAM 



SUPERHEAT, °F 


40 


90 


200 


300 


400 


500 


600 


Absolute Pressure 


' t = 141.7 


191.7 


301.7 


401.7 


501.7 


601.7 


701.7 


Lbs. per Square Inch 


V = 357.8 


387.9 


453.7 


513.4 


573.1 


632.7 


692.4 


1 


' //= 1122.6 


1145.3 


1195.6 


1241.5 


1287.6 


1334.1 


1381.0 




n = 2.0069 


2.0434 


2.1145 


2.1701 


2.2218 


2.2679 


2.4100 




' t = 166.1 


216.1 


326.1 


426.1 


526.1 


626.1 


726.1 




V = 186.1 


201.2 


234.2 


264.1 


293.9 


323.8 


353.6 


2 


< H= 1133.2 


1156.1 


1206.4 


1252.4 


1298.6 


1345.2 


1392.2 




n = 1.9486 


1.9836 


2.0529 


2.1071 


2.1586 


2.2044 


2.2459 




' t = 280.1 


330.1 


440.1 


540.1 


640.1 


740.1 


840.1 




V = 17.35 


18.61 


21.32 


23.77 


26.20 


28.61 


31.01 


35 


* H= 1179.6 


1203.4 


1255.6 


1302.8 


1350.1 


1397.5 


1445.4 




n = 1.7402 


1.7712 


1.8330 


1.8827 


1.9277 


1.9688 


2.0078 




' t = 367.8 


417.8 


527.8 


627.8 


727.8 


827.8 


927.8 


100 


V = 4.72 


5.07 


5.80 


6.44 


7.07 


7.69 


8.31 


< H= 1208.4 


1234.6 


1289.4 


1337.8 


1385.9 


1434.1 


1482.5 




n = 1.6294 


1.6600 


1.7188 


1.7656 


1.8079 


1.8468 


1.8829 




' t = 393.1 


443.1 


553.1 


653.1 


753.1 


853.1 


953.1 


140 


F=3.44 


3.70 


4.24 


4.71 


5.16 


5.61 


6.06 


' H = 1215.8 


1242.8 


1298.2 


1346.9 


1395.4 


1443.8 


1492.4 




n = 1.6031 


1.6338 


1.6916 


1.7376 


1.7792 


1.8177 


1.8533 




' t = 398.5 


448.5 


558.5 


658.5 


758.5 


858.5 


958.5 


150 


F=3.22 


3.46 


3.97 


4.41 


4.84 


5.25 


5.67 


' H= 1217.3 


1244.4 


1300.0 


1348.8 


1397.4 


1445.9 


1494.6 




n = 1.5978 


1.6286 


1.6862 


1.7320 


1.7735 


1.8118 


1.8474 



t = temperature Fahrenheit ; V = specific volume ; H = total heat above 32° F. ; 
n — entropy above 32° F. 

(Condensed from Steam Tables and Diagrams, by Marks and Davis, with the per- 
mission of the publishers, Messrs. Longmans, Green, & Co.) 



THEORY OF VAPORS 



247 



PROPERTIES OF DRY SATURATED STEAM 

(Condensed from Steam Tables and Diagrams, by Marks and Davis, with the permis- 
sion of the publishers, Messrs. Longmans, Green, & Co.) 



p 


T 


V 


Ji 


L 


H 


r 


n w 


n e 


«s 


1 


101.83 


333.0 


69.8 


1034.6 


1104.4 


972.9 


0.1327 


1.8427 


1.9754 


2 


126.15 


173.5 


94.0 


1021.0 


1115.0 


956.7 


0.1749 


1.7431 


1.9180 


3 


141.52 


118.5 


109.4 


1012.3 


1121.6 


946.4 


0.2008 


1.6840 


1.8848 


4 


153.01 


90.5 


120.9 


1005.7 


1126.5 


938.6 


0.2198 


1.6416 


1.8614 


5 


162.28 


73.33 


130.1 


1000.3 


1130.5 


932.4 


0.2348 


1.6084 


1.8432 


6 


170.06 


61.89 


137.9 


995.8 


1133.7 


927.0 


0.2471 


1.5814 


1.8285 


7 


176.85 


53.56 


144.7 


991.8 


1136.5 


922.4 


0.2579 


1.5582 


1.8161 


8 


182.86 


47.27 


150.8 


988.2 


1139.0 


918.2 


0.2673 


1.5380 


1.8053 


9 


188.27 


42.36 


156.2 


985.0 


1141.1 


914.4 


0.2756 


1.5202 


1.7958 


10 


193.22 


38.38 


161.1 


982.0 


1143.1 


910.9 


0.2832 


1.5042 


1.7874 


11 


197.75 


35.10 


165.7 


979.2 


1144.9 


907.8 


0.2902 


1.4895 


1.7797 


12 


201.96 


32.36 


169.9 


976.6 


1146.5 


904.8 


0.2967 


1.4760 


1.7727 


13 


205.87 


30.03 


173.8 


974.2 


1148.0 


902.0 


0.3025 


1.4639 


1.7664 


14 


209.55 


28.02 


177.5 


971.9 


1149.4 


899.3 


0.3081 


1.4523 


1.7604 


15 


213.0 


26.27 


181.0 


969.7 


1150.7 


896.8 


0.3133 


1.4416 


1.7549 


16 


216.3 


24.79 


184.4 


967.6 


1152.0 


894.4 


0.3183 


1.4311 


1.7494 


17 


219.4 


23.38 


187.5 


965.6 


1153.1 


892.1 


0.3229 


1.4215 


1.7444 


18 


222.4 


22.16 


190.5 


963.7 


1154.2 


889.9 


0.3273 


1.4127 


1.7400 


19 


225.2 


21.07 


193.4 


961.8 


1155.2 


887.8 


0.3315 


1.4045 


1.7360 


20 


228.0 


20.08 


196.1 


960.0 


1156.2 


885.8 


0.3355 


1.3965 


1.7320 


21 


230.6 


19.18 


198.8 


958.3 


1157.1 


883.9 


0.3393 


1.3887 


1.7280 


22 


233.1 


18.37 


201.3 


956.7 


1158.0 


882.0 


0.3430 


1.3811 


1.7241 


23 


235.5 


17.62 


203.8 


955.1 


1158.8 


880.2 


0.3465 


1.3739 


1.7204 


24 


237.8 


16.93 


206.1 


953.5 


1159.6 


878.5 


0.3499 


1.3670 


1.7169 


25 


240.1 


16.30 


208.4 


952.0 


1160.4 


876.8 


0.3532 


1.3604 


1.7136 


26 


242.2 


15.72 


210.6 


950.6 


1161.2 


875.1 


0.3564 


1 .3542 


1.7106 


27 


244.4 


15.18 


212.7 


949.2 


1161.9 


873.5 


0.3594 


1.3483 


1.7077 


28 " 


246.4 


14.67 


214.8 


947.8 


1162.6 


872.0 


0.3623 


1.3425 


1.7048 


29 


248.4 


14.19 


216.8 


946.4 


1163.2 


870.5 


0.3652 


1.3367 


1.7019 


30 


250.3 


13.74 


218.8 


945.1 


1163.9 


869.0 


0.3680 


1.3311 


1.6991 


31 


252.2 


13.32 


220.7 


943.8 


1164.5 


867.6 


0.3707 


1.3257 


1.6964 


32 


254.1 


12.93 


222.6 


942.5 


1165.1 


866.2 


0.3733 


1.3205 


1.6938 


33 


255.8 


12.57 


224.4 


941.3 


1165.7 


864.8 


0.3759 


1.3155 


1.6914 


34 


257.6 


12.22 


226.2 


940.1 


1166.3 


863.4 


0.3784 


1.3107 


1.6891 


35 


259.3 


11.89 


227.9 


938.9 


1166.8 


862.1 


0.3808 


1.3060 


1.6868 


36 


261.0 


11.58 


229.6 


937.7 


1167.3 


860.8 


0.3832 


1.3014 


1.6846 


37 


262.6 


11.29 


231.3 


936.6 


1167.8 


859.5 


0.3855 


1.2969 


1.6824 


38 


264.2 


11.01 


232.9 


935.5 


1168.4 


858.3 


0.3877 


1.2925 


1.6802 


39 


265.8 


10.74 


234.5 


934.4 


1168.9 


857.1 


0.3899 


1.2882 


1.6781 


40 


267.3 


10.49 


236.1 


933.3 


1169.4 


855.9 


0.3920 


1.2841 


1.6761 


41 


268.7 


10.25 


237.6 


932.2 


1169.8 


854.7 


0.3941 


1.2800 


1 6741 


42 


270.2 


10.02 


239.1 


931.2 


1170.3 


853.6 


0.3962 


1.2759 


1.6721 


43 


271.7 


9.80 


240.5 


930.2 


1170.7 


852.4 


0.3982 


1.2720 


1.6702 


44 


273.1 


9.59 


242.0 


929.2 


1171.2 


851.3 


0.4002 


1.2681 


1.6683 


45 


274.5 


9.39 


243.4 


928.2 


1171.6 


850.3 


0.4021 


1.2644 


1.6665 


46 


275.8 


9.20 


244.8 


927.2 


1172.0 


849.2 


0.4040 


1.2607 


1.6647 


47 


277.2 


9.02 


246.1 


923.3 


1172.4 


848.1 


0.4059 


1.2571 


1.6630 


48 


278.5 


8.84 


247.5 


925.3 


1172.8 


847.1 


0.4077 


1.2536 


1.6613 


49 


279.8 


8.67 


248.8 


924.4 


1173.2 


846.1 


0.4095 


1.2502 


1.6597 


50 


281.0 


8.51 


250.1 


923.5 


1173.6 


845.0 


0.4113 


1.2468 


1.6581 



248 



APPLIED THERMODYNAMICS 



PROPERTIES OF DRY SATURATED STEAM — Continued 

(Condensed from Steam Tables and Diagrams, by Marks and Davis, with the permis- 
sion of the publishers, Messrs. Longmans, Green, & Co.) 



p 


T 


V 


h 


L 


IT 


r 


n w 


n e 


n s 


51 


282.3 


8.35 


251.4 


922.6 


1174.0 


844.0 


0.4130 


1.2435 


1.6565 


52 


283.5 


8.20 


252.6 


921.7 


1174.3 


843.1 


0.4147 


1.2402 


1.6549 


53 


284.7 


8.05 


253.9 


920.8 


1174.7 


842.1 


0.4164 


1.2370 


1.6534 


54 


285.9 


7.91 


255.1 


919.9 


1175.0 


841.1 


0.4180 


1.2339 


1.6519 


55 


287.1 


7.78 


256.3 


919.0 


1175.4 


840.2 


0.4196 


1.2309 


1.6505 


56 


288.2 


7.65 


257.5 


918.2 


1175.7 


839.3 


0.4212 


1.2278 


1.6490 


57 


289.4 


7.52 


258.7 


917.4 


1176.0 


838.3 


0.4227 


1.2248 


1.6475 


58 


290.5 


7.40 


259.8 


916.5 


1176.4 


837.4 


0.4242 


1.2218 


1.6460 


59 


291.6 


7.28 


261.0 


915.7 


1176.7 


836.5 


0.4257 


1.2189 


1.6446 


60 


292.7 


7.17 


262.1 


914.9 


1177.0 


835.6 


0.4272 


1.2160 


1.6432 


61 


293.8 


7.06 


263.2 


914.1 


1177.3 


834.8 


0.4287 


1.2132 


1.6419 


62 


294.9 


6.95 


264.3 


913.3 


1177.6 


833.9 


0.4302 


1.2104 


1.6406 


63 


295.9 


6.85 


265.4 


912.5 


1177.9 


833.1 


0.4316 


1.2077 


1.6393 


64 


297.0 


6.75 


266.4 


911.8 


1178.2 


832.2 


0.4330 


1.2050 


1.6380 


65 


298.0 


6.65 


267.5 


911.0 


1178.5 


831.4 


0.4344 


1.2034 


1.6368 


66 


299.0 


6.56 


268.5 


910.2 


1178.8 


830.5 


0.4358 


1.2007 


1.6355 


67 


300.0 


6.47 


269.6 


909.5 


1179.0 


829.7 


0.4371 


1.1972 


1.6343 


68 


301.0 


6.38 


270.6 


908.7 


1179.3 


828.9 


0.4385 


1.1946 


1.6331 


69 


302.0 


6.29 


271.6 


908.0 


1179.6 


828.1 


0.4398 


1.1921 


1.6319 


70 


302.9 


6.20 


272.6 


907.2 


1179.8 


827.3 


0.4411 


1.1896 


1.6307 


71 


303.9 


6.12 


273.6 


906.5 


1180.1 


826.5 


0.4424 


1.1872 


1.6296 


72 


304.8 


6.04 


274.5 


905.8 


1180.4 


825.8 


0.4437 


1.1848 


1.6285 


73 


305.8 


5.96 


275.5 


905.1 


1180.6 


825.0 


0.4449 


1.1825 


1.6274 


74 


306.7 


5.89 


276.5 


904.4 


1180.9 


824.2 


0.4462 


1.1801 


1.6263 


75 


307.6 


5.81 


277.4 


903.7 


1181.1 


823.5 


0.4474 


1.1778 


1.6252 


80 


312.0 


5.47 


282.0 


900.3 


1182.3 


819.8 


0.4535 


1.1665 


1.6200 


85 


316.3 


5.16 


286.3 


897.1 


1183.4 


816.3 


0.4590 


1.1561 


1.6151 


90 


320.3 


4.89 


290.5 


893.9 


1184.4 


813.0 


0.4644 


1.1461 


1.6105 


95 


324.1 


4.65 


294.5 


890.9 


1185.4 


809.7 


0.4694 


1.1367 


1.6061 


100 


327.8 


4.429 


298.3 


888.0 


1186.3 


806.6 


0.4743 


1.1277 


1.6020 


105 


331.4 


4.230 


302.0 


885.2 


1187.2 


803.6 


0.4789 


1.1191 


1.5980 


110 


334.8 


4.047 


305.5 


882.5 


1188.0 


800.7 


0.4834 


1.1108 


1.5942 


115 


338.1 


3.880 


309.0 


879.8 


1188.8 


797.9 


0.4877 


1.1030 


1.5907 


120 


341.3 


3.726 


312.3 


877.2 


1189.6 


795.2 


0.4919 


1.0954 


1.5873 


125 


344.4 


3.583 


315.5 


874.7 


1190.3 


792.6 


0.4959 


1.0880 


1.5839 


130 


347.4 


3.452 


318.6 


872.3 


1191.0 


790.0 


0.4998 


1.0809 


1.5807 


140 


353.1 


3.219 


324.6 


867.6 


1192.2 


785.0 


0.5072 


1.0675 


1.5747 


150 


358.5 


3.012 


330.2 


863.2 


1193.4 


780.4 


0.5142 


1.0550 


1.5692 


160 


363.6 


2.834 


335.6 


858.8 


1194.5 


775.8 


0.5208 


1.0431 


1.5639 


170 


368.5 


2.675 


340.7 


854.7 


1195.4 


771.5 


0.5269 


1.0321 


1.5590 


180 


373.1 


2.533 


345.6 


850.8 


1196.4 


767.4 


0.5328 


1.0215 


1.5543 


190 


377.6 


2.406 


350.4 


846.9 


1197.3 


763.4 


0.5384 


1.0114 


1.5498 


200 


381.9 


2.290 


354.9 


843.2 


1198.1 


759.5 


0.5437 


1.0019 


1.5456 


210 


386.0 


2.187 


359.2 


839.6 


1198.8 


755.8 


0.5488 


0.9928 


1.5416 


220 


389.9 


2.091 


363.4 


836.2 


1199.6 


752.3 


0.5538 


0.9841 


1.5379 


230 


393.8 


2.004 


367.5 


832.8 


1200.2 


748.8 


0.5586 


0.9758 


1.5344 


240 


397.4 


1.924 


371.4 


829.5 


1200.9 


745.4 


0.5633 


0.9676 


1.5309 


250 


401.1 


1.850 


375.2 


82.3.3 


1201.5 


742.0 


0.5676 


0.9600 


1.5276 



THEORY OF VAPORS 249 

(1) Phil. Trans., 1854, CXLIV, 360. (2) Phil. Trans., 1854, 336; 1862, 579. 
(3) Theorie Mecanique de la Chaleur, 2d ed., I, 195. (4) Wood, Thermodynamics, 
1905, 396. (5) Wiedemann, Ann. der Phys. und Chem., 1880, Vol. IX. (6) Technical 
Thermodynamics (Klein), 1907, II, 215. (7) Mitteilungen uber Forschungsarbeiten, 
etc., 21 ; 33. (8) Peabody, Steam Tables, 1908, 9 ; Marks and Davis, Tables and 
Diagrams, 1909, 88; Phil. Trans., 199 A (1902), 149-263. (9) The Steam Engine, 
1897, 601. (10) Op. cit., II, App. XXX. (11) The Bichards Steam Engine Indica- 
tor, by Charles T. Porter. (12) Trans. A. S. 31. E., XI. (13) Dubois ed., II, ii, 1884. 
(14) Peabody, op. cit. (15) Trans. A S. 31. E., XII, 590. (16) Ann. der Physik, 4, 
26, 1908, 833. (17) Trans. A. S. 31. E., XXX, 1419-1432. (18) Tables and Diagrams 
of The Thermal Properties of Saturated and Superheated Steam, 1909. (19) Zeits. 
fur Instrumentenlcunde, XIII, 329. (20) Wissenschaftliche Abhandlungen, III, 71. 

(21) Sitzungsberichte K. A. W. in Wien, Math.-natur Klasse, CVII, II, Oct., 1899. 

(22) hoc. cit., note (7). (23) Op. cit., 417. (24) Comptes Bendus, LXII, 56 ; Bull, 
de la Soc. Industr. de 31ulhouse, CXXXIII, 129. (25) Boulvin's method : see Berry, 
Tlie Temperature Entropy Diagram, 1906, 34. (26) Zeuner, op. cit., II, 207-208. 
(27) Nichols and Franklin, Elements of Physics, I, 194. (28) Phil. Trans., 1869, II, 
575. (29) Zeits. Yer. Deutsch. Ing., 1904, 24. (30) Trans. A. S. 31. E., XXVIII, 8, 
1264. (31) Ann. der Phys., Leipzig, 1905, IV, XVIII, 739. (32) Zeits. Ver. Deutsch. 
Ing., Oct. 19, 1907. (33) Mitteil. uber For schungsarb., XXXVI, 109. (34) Trans. 
A. S. 31. E., XXVIII, 10, 1695. (35) Trans. A. S. 31. E., XXIX, 6, 633. (36) Ibid., 
XXX, 5, 533. (37) Ibid., XXX, 9, 1227. (38) Op. cit., II, 239. (39) Pea- 
body, Op. cit., 111. (40) The Steam Engine, 1905, 68. (41) Trans. A. S. 
31. E., XXIX, 6. (42) Op. cit., II, Apps. XXXIV, XXXV, XL, XLIV, XLII, 
XXXVIII. (43) Op. cit., 407 et. seq. (44) Op. cit, 600. (45) Comptes Bendus, 
CII, 1886, 1202. (46) Ibid., CXIV, 1892, 1093; CXIII, 1891. (47) Zeits. fur die 
gesamte Kalte-Industrie, 1895, 66-85. (48) Op. cit., II, App. L. (49) Machines a 
froid, Paris, 1878. (50) Elleau and Ennis, Jour. Frank. Inst., Mar., Apr., 1898 ; 
Dietrici, Zeits. Kalte-Ind., 1904. (51) Op. cit., II, App. XLVI. (52) Zeits. fur die 
gesamte Kalte-Industrie, 1904. The heavy line across the table on page 422 indicates 
a break in continuity between the two sources of data. The same break is responsible 
for the notable irregularity in the saturation and constant dryness curves on the ammonia 
entropy diagram, Fig. 316. (53) Tables of the Properties of Saturated Steam and other 

Yapors, 1890. (54) See Jacobus, Trans. A. S. 31. E., XII. (55) Jour. Frank. Inst., 
Dec, 1890. (56) Op. cit., 466. (57) Mem. de VInstitut de France, XXI, XXVI. 
(58) Landolt and Bornstein, Physikalische-chemische Tab ell en ; Gmelin ; Peabody, 
Thermodynamics, 118. (59) Andre'eff, Ann. Chem. Pharm., 1859. (60) Trans. 
A. S. 31. E., XXV, 176. (61) Tables, etc., 1890. (62) Comptes Bendus, CXIX, 
1894, 404-407. (63) Op. cit., App. XL VIII. (64) Op. cit., 468. (65) Trans. 
A. S. M. E., XII. (66) Op. cit., II, App. XXXII. (67) Trans. A. S. M. E., XXI, 
3, 406. (68) Wied. Annallen, (4), XVI, 1905, 593-620. (69) Wied. Annallen, (4), 
XVIII, 1905, 739-756 ; (4), XXIII, 1907, 809-845. (70) Marks and Davis, op. cit., 95. 
(71) Trans. A. S. M. E., May, 1908. 

SYNOPSIS OF CHAPTER XII 

The temperature remains constant during evaporation ; that of the liquid is the same 
as that of the vapor; increase of pressure raises the boiling point, and vice versa; 
it also increases the density. There is a definite boiling point for each pressure. 

Saturated vapor is vapor at minimum temperature and maximum density for the given 
pressure. 

Superheated vapor is an imperfect gas, produced by adding heat to a dry saturated vapor. 



250 APPLIED THERMODYNAMICS 



Saturated Steam 

The principal effects of heat are, h = t — 32, e = ' ~ ' , 

778 

r = Z-e, iT^ft + L = h + r + e. 
As p increases, t, h, e and H increase, and r and L decrease. 

H= H 212 + 0.3745(£ - 212) - 0.00055(* - 212)2. 

Factor of evaporation = L + ( h ~ h() )' 
J F 970.4 

The pressure increases more rapidly than the temperature. 

Characteristic equation for steam, pv = aT — p(l + op) (- d 

Saturated steam may be dry or wet. For wet steam, 

h — ho, L = xLo, H = xLq + ho, r = xr , e = ice , 
and the factor of evaporation is xL +( h ~ h o) m The vo i ume j s w = V+x( Wo- V) . 

The water line shows the volume of water at various temperatures ; the saturation curve 
shows the relation between volume and temperature of saturated steam. Approxi- 
mately, pv T * = constant. The isothermal is a line of constant pressure. 

The path during evaporation is (a) along the water line (6) across to the saturation 
curve at constant pressure and temperature. If superheating occurs, the path pro- 
ceeds at constant pressure and increasing temperature to the right of the satura- 
tion curve. 

T 

On the entropy diagram, the equation of the water line is n = clog e — . The distance 

between the water line and the saturation curve is JV = — . Constant dryness 

curves divide this distance in equal proportions. Lines of constant total heat may 
be drawn. The specific heat of steam kept dry is negative. The dryness changes 
during adiabatic expansion. The temperature of inversion is that temperature at 
which the specific heat of saturated steam is zero. The change of internal energy 
and the external work along any path of saturated steam may be represented on the 
entropy diagram. 

w= v mzdT 

T dP 

Constant volume lines may be plotted on the entropy diagram, permitting of the trans- 
fer of any point or path from the PVto the TiY plane. The temperature after 
expansion at contant entropy to a limiting volume can best be obtained from the 
entropy diagram. 

The critical temperature is that temperature at which the latent heat becomes zero 
{689° F.). 

Saturated vapor (dry or wet), superheated vapor, gas ; physical states in relation to the 
critical temperature ; shape of isothermals. 

The isodynamic path for saturated steam touches the saturation curve at one point 
only. 



THEORY OF VAPORS 251 

Superheated Steam 

The specific heat has been in doubt. Its value increases with the pressure, and varies 
with the temperature. 

H = H + k v (T-t). fe= T *~ h ' Bb-S e =-k Pl (T e -T b -) + k P2 (T d -T e ). 

ftp! -I — t 

Factor of evaporation = &*«t + k p (T - t) - h pv= 64901 T _ 22 .5819 P°™. 

o i 0.4 

PV= 0.594 T- 0.00178 P. B = ± 85.8. y = ± 1.298. 

Paths of Vapors 

Adiabatic equation : — = c log e — | Approximately, PV n = constant. Values of n. 

External work along an adiabatic = h— H + xr — XB. 
Continuously superheated adiabatic, e.g., 

log « ioT7 + I + fcl loge f = loge z^h + - + * 2 log ^-- 

491.6 £ £ 491.6 m 2* 

Adiabatic crossing the saturation curve : 

T 

Method of drawing constant pressure lines on the entropy diagram : n = &plog e — 

Method of drawing lines of constant total heat. 

Use of the entropy diagram for graphically solving problems : dryness after expansion ; 

work done during expansion ; mixing ; heat contents. 
The Mollier coordinates, total heat and entropy. The total heat-pressure diagram. 

Vapors in General 

dT 778 dT dT T T dP 

When the pressure-temperature relation and the characteristic equation are given, we 
may compute L for various temperatures, and the specific heat of the vapor. 

Vapors in engineering, Ammonia : log j) = 8.4079-^, ^=91 — ^~, fc=0.508, 

vapor density = 0.597 (air = 1), specific volume of liquid = 0.025, its specific heat 
= 1.02. Sulphur dioxide: k = 0.15438, vapor density = 2.23, specific volume of 
liquid = 0.0007, its specific heat = 0.4. PV= 26.4 T— 184 P - 22 . Pressure-tem- 
perature relation. L = 176 — 0.27 (t— 32). 

Steam Cycles 

Efficiency = work done -^ gross heat absorbed. 

The Car not cycle is impracticable : the steam power plant operates in the Clausius cycle. 

(T -t)(l+^-tlog e I 



Efficiency of Clausius cycle =- 



T-t+ XL 



252 APPLIED THERMODYNAMICS 

Bankine cycle (incomplete expansion) — determination Of efficiency, with steam 
initially wet or dry. 

Non-expansive cycle : efficiency = (Pb-P)(™ b - 0.017) ^ 

h e — hd + xLi 

1433 log e --0. 695 (T-£) 

Tambour cycle : steam dry during expansion ; efficiency— rp ; 

L f + 1433 log e — - 0.695 ( T- t) 

computation of heat supplied by jacket. 

Superheated cycle : efficiency is increased if the final dryness is properly adjusted and 

the ratio of expansion is not too low. 
Numerical comparison of seventeen cycles for efficiency and capacity : steam should 

he initially dry. The ratio of expansion should be large for efficiency and small 

for capacity. 

The Steam Tables 

Computation is from p (ox t) to t (or p), H, h, L, -^, V, e, r, n w , n e , n s . 

dt 
The superheated tables give n, F", H, t, for various superheats at various pressures ; all 
values depending on H sat , n sat , and k p . 

PROBLEMS 

Note. Problems not marked T are to be solved without the use of the steam 
table. In all cases where possible, computed results should be checked step by step 
with those read from the three charts, Figs. 175, 177, 185. 

T 1. The weight per cubic foot of water at 32° F. being 62.42, and at 250.3° F., 
58.84, compute in heat units the external work done in heating one pound of water at 
pressure from 32° to 250.3°. (The pressure is that of saturated steam at a temperature 
of 250.3°.) 

2. Forp = 100, t = 327.8°, W= 4.429, compute h (approximately), H, L, e, r in 
the order given. Why do not the results agree with those in the table ? 

TZ. Find the factor of evaporation for dry steam at 95 lb. pressure, the feed- 
water temperature being 153° F. 

T 4. Given the formula, log p — c — — , T being the absolute tempera- 
ture and p the pressure per square foot, find the value of S- for p = 100 lb. per square 

dt 

inch, t — 327.8° F. Check roughly by observing nearest differences in the steam table. 

T 5. What increase in steam pressure accompanies an increase in temperature 
from 353.1° F. to 393.8° F. ? Compare the percentages of increase of absolute pressure 
and absolute temperature. 

T6. Find the values of the constants in the Rankine and Zeuner equations (Art. 
363), at 100 lb. pressure. 

T7. From Art. 363, find the volume of dry steam at 240.1° F. in four ways. 
Compare with the value given in the steam table and explain the disagreement. 

8. At 100 lb. pressure, the latent heat per pound is 888.0 ; per cubic foot, it is 
200.3. Find the specific volume. 



PROBLEMS 253 

9. For the conditions given in Problem 2, W being the volume of dry steam, find 
the five required thermal properties of steam 95 per cent dry. Find its volume. 

T 10. State the condition of steam (wet, dry, or superheated) when (a) p = 100, 
t = 327.8 ; (b) p = 95, v = 4.0 ; (c) p = 80, t = 360. 

11. Determine the path on the entropy diagram for heating from 200° to 240° F. 
a fluid the specific heat of which is 1.00 + at, in which t is the Fahrenheit temperature 
and a = 0.0044. 

T 12. Find the increases in entropy during evaporation to dry steam at the fol- 
lowing temperatures : 228°, 261°, 386° F. 

P13. Compute from Art. 368 the specific volume of dry steam at 327.8 F. What 
is its volume if 4 per cent wet ? (See Problem 4.) 

T 14. Find the entropy, measured from 32° F., of steam at 327.8° F., 65 per cent 
dry, (a) by direct computation, (b) from the steam table. Explain any discrepancy. 

T 15. Dry steam at 100 lb. pressure is compressed without change of internal 
energy until its pressure is 200 lb. Find its dryness after compression. 

T 16. Find the dryness of steam at 300° F. if the total heat is 800 B. t. u. 

T 17. Find the entropy of steam at 130 lb. pressure when the total heat is 840 B. t. u. 

T 18. One pound of steam at 300° F., having a total heat of 800 B. t. u., expands 
adiabatically to 1 lb. pressure. Find its dryness, entropy, and total heat after expan- 
sion. What weight of steam was condensed during expansion ? 

19. Transfer a wet steam adiabatic from the TN to the PV plane, by the graphi- 
cal method. 

20. Transfer a constant dryness line in the same manner. 

21. Sketch on the TJY and PV planes the saturation curve and the water line in 
the region of the critical temperature. 

T 22. At what stage of dryness, at 300° F., is the internal energy of steam equal 
to that of dry steam at 228° F ? 

T23. At what specific volume, at 300° F., is the internal energy of steam equal 
to that of dry steam at 228° F. ? 

P24. Compute from the Thomas experiments the total heat in steam at 100 lb. 
pressure and 440° F. 

T 25. Find the factor of evaporation for steam at 100 lb. pressure and 500° F. from 
feed water at 153° F. 

P26. In Problem 18, find the volume after expansion, and compare with the vol- 
ume that would have been obtained by the use of Zeuner's exponent (Art. 394). 
Which result is to be preferred ? 

T27. Using the Knoblauch and Jacob values for the specific heat, and determin- 
ing the initial properties in at least five steps, compute the initial entropy and total 
heat and the condition of steam after adiabatic expansion from P = 100, T= 700° F.. 
to p = 13. Find its volume from the formula in Art. 390. Compare with the volume 
given by the equation py 1 - 298 =pv 1 - 298 . (Assume that the superheated table shows 
the steam to be superheated about 55° F. at the end of expansion.) 

T 28. Compute the dryness of steam after adiabatic expansion from P = 140, 
T = 753.1° F., to t = 153° F. Find the change in volume during expansion. 



254 APPLIED THERMODYNAMICS 

T 29. Find the external work done in Problems 27 and 28, along the expansive 
paths. 

T 30. At what temperature is the total heat in steam at 100 lb. pressure 1200 B. t. u. ? 
31. Find the efficiency of the Carnot cycle between 341.3° F. and 101.83° F. 

T 32. Find the efficiency of the Clausius cycle, using initially dry steam between 
the same temperature limits. 

T 33. In Problem 32, find the efficiency if the steam is initially 60 per cent dry. 

T 34. In Problem 32, find the efficiency if expansion terminates when the volume 
is 12 cu. ft. (Rankine cycle). 

7^35. In Problem 32, find the efficiency if there is no expansion. 

T 36. Find the efficiency of the Pambour cycle between the temperature limits 
given in Problem 31. How much heat is supplied by the jacket ? 

T37. Find the efficiency of this Pambour cycle if expansion terminates when the 
volume is 12 cu. ft. 

r38. Steam initially at 140 lb. pressure and 443.1° F. is worked (a) in the Clau- 
sius cycle, (6) in the Rankine cycle, with the same ratio of expansion as in Problem 
37. Find the efficiency in each case, the lower temperature being 101.83° F. Find the 
efficiency of the Rankine cycle in which the maximum volume is 5 cu. ft. (See foot- 
note, Case VIII, Art. 417.) 

T 39. At what per cent of dryness is the volume of steam at 100 lb. pressure 
3 cu. ft. ? 

T40. Steam at 100 lb. pressure is superheated so that adiabatic expansion to 
261° F. will make it just dry. Find its condition if adiabatic expansion is then carried 
on to 213° F. Find the external work done during the whole expansion. 

T41. Steam passes adiabatically through an orifice, the pressure falling from 140 
to 100 lb. When the inlet temperature of the steam is 500° F., its outlet temperature 
is 494° F. ; and when the inlet temperature is 600° F. , the outlet temperature is 595° F. 
The mean value of the specific heat at 140 lb. pressure between 600° F. and 500° F. is 
0.498. Find the mean value at 100 lb. pressure between 595° F. and 494° F. How 
does this value agree with that found by Knoblauch and Jacob ? 

T42. Find from Problem 41 and Fig. 171 the total heat in saturated steam at 140 
lb. pressure, in two ways, that at 100 lb. pressure being 1186.3. 

T43. Plot on a total heat-pressure diagram the saturation curve, the constant 
dryness curve for x = 0.85, the constant temperature curve for T=500° F., and a 
constant volume curve for V= 13, passing through both the wet and the superheated 
regions. Use a vertical pressure scale of 1 in. = 20 lb., and a horizontal heat scale of 
1 in. = 20 B. t. u. 

44. Compute the temperature of inversion of ammonia, given the equation, 
L = 555.5 - 0.613 T° F., the specific heat of the liquid being 1.0. What is the result 
if L = 555.5 - 0.613 T- 0.000219 T* (Art. 401)? 

45. Compute the pressure of the saturated vapor of sulphur dioxide at 60° F. (Art. 
404). (Compare Table, page 424.) 

746. Compare the capacities of the cycles in Problems 31-37, as in Art. 418. 

47. Sketch the water line, the saturation curve, an adiabatic for saturated steam, 
and a constant dryness line on the PT plane. 



PROBLEMS 255 

T48. A 10-gal. vessel contains 0.1 lb. of water and 0.7 lb. of dry steam. What is 
the pressure ? 

T49. A cylinder contains 0.25 lb. of wet steam at 58 lb. pressure, the volume 
being 1.3 cu. ft. What is the quality of the steam ? 

T 50. What is the internal energy of the substance in the cylinder in Problem 49 ? 

T51. Steam at 140 lb. pressure, superheated 400° F., expands adiabatically until 
its pressure is 5 lb. Find its final quality and the ratio of expansion. 

T 52. The same steam expands adiabatically until its dryness is 0.98. Find its 
pressure. 

T 53. * The same steam expands adiabatically until its specific volume is 50. Find 
its pressure and quality. 

T54. Steam at 200 lb. pressure, 94 per cent dry, is throttled as in Art. 387. At 
what pressure must the throttle valve be set to discharge dry saturated steam ? 

T 55. Steam is throttled from 200 lb. pressure to 15 lb. pressure, its temperature 
becoming 235.5° F. What was its initial quality ? (Use Fig. 175.) 

56. Represent on the entropy diagram the factor of evaporation of superheated 
steam. 

57. Check by accurate computations all the values given in the saturated steam 
table for t = 180° F., using — 459.64° F. for the absolute zero, 14.696 lb. per square 
inch for the standard atmosphere, 777.52 for the mechanical equivalent of heat, and 
0.017 as the specific volume of water. Use Thiesen's formula for the pressure : 

(t + 459.6) log -J2- = 5.409 (t - 212) - 8.71 x 10" 10 [(689 - t)* - 477 4 ] ; 

t being the Fahrenheit temperature and p the pressure in pounds per square inch. Use 
the Knoblauch, Linde and Klebe formula for the volume and the Davis formula for 
the total heat. Compute the entropy and heat of the liquid in eight steps, using the 
following values for the specific heat of the liquid : 

at 40°, 1.0045 ; at 120°, 0.9974 ; 

at 60°, 0.9991 ; at 140°, 0.9987 ; 

at 80°, 0.997 ; at 160°, 1.0002 ; 

at 100°, 0.99675 ; at 180°, 1.0020. 

Explain the reasons for any discrepancies. 

T 58. Check the properties given in the superheated steam table for P = 25 with 
200° of superheat, using Knoblauch values for the specific heat, in at least three steps, 
and using the Knoblauch, Linde and Klebe formula for the volume. Explain any 
discrepancies. 

59. Represent on the entropy diagram the temperature of inversion of a dry vapor. 

* This is typical of a class of problems the solution of which is difficult or impos- 
sible without plotting the properties on charts like those of Figs. 175, 177, 185. Prob- 
lem 53 may be solved by a careful inspection of the total heat-pressure and Mollier 
diagrams, with reasonable accuracy. The approximate analytical solution will be found 
an interesting exercise. We have no direct formula for relation between V and T, 
although one may be derived by combining the equations of Rankine or Zeuner (Art 
with that in Problem 4. 



CHAPTER XIII 

THE STEAM ENGINE 
Practical Modifications of the Rankine Cycle 

422. The Steam Engine. Figure 186 shows the working parts. 
The piston P moves in the cylinder A, communicating its motion 
through the piston rod R, crosshead (7, and connecting rod M to the 
disk crank D on the shaft #, and thus to the belt wheel W. The 
guides on which the crosshead moves are indicated by 6r, H, the 
frame which supports the working parts by F. Journal bearings 
at B and support the shaft. The function of the mechanism is to 
transform the to-and-fro rectilinear motion of the piston to a rotatory 
movement at the crank. Without entering into details at this point, 
it may be noted that the valve V, which alternately admits of the 
passage of steam through either of the ports X, Z", is actuated by a 
valve rod /traveling from a rocker </, which derives its motion from 
the eccentric rod iVand the eccentric E. In the end view, L is the 
opening for the admission of steam to the steam chest K, Q is a sim- 
ilar opening for the exit of the steam (shown also in the plan), and 
Fis the valve. 

423. The Cycle. With the piston in the position shown, and 
moving to the left, steam is passing from the steam chest through Y 
into the cylinder, while another mass of steam, which has expended 
its energy, is passing from the other side of the piston through the 
port X and the opening Q to the atmosphere or the condenser. 
When the piston shall have reached its extreme left-hand position, 
the valve will have moved to the right, the port J 7 ' will have been 
cut off from communication with K, and the steam on the right of 
the piston will be passing through Y to Q. At the same time the 
port X will be cut off from Q and placed in communication with K. 
The piston then makes a stroke to the right, while the valve moves 
to the left. The engine shown is thus double-acting. 

256 



THE STEAM ENGINE 



257 





258 



APPLIED THERMODYNAMICS 



If the valve moved instantaneously from one position to the other 
precisely at the end of the stroke, the PV diagram representing 
the changes in the fluid on either side of the piston would resemble 
ebtd, Fig. 184. Along eb, the steam would be passing from the 
steam chest to the cylinder, the pressure being practically constant 
because of the comparatively enormous storage space in the boiler, 
while the piston moved outward, doing work. At 5, the supply of 
steam would cease, while communication would be immediately 
opened with the atmosphere or the condenser, causing the fall of 
pressure along bt. The piston would then make its return stroke, 
the steam passing out of the cylinder at practically constant pressure 
along td, and at d the position of the valve would again be changed, 
closing the exhaust and opening the supply and giving the instan- 
taneous rise of pressure indicated by de. 

424. Expansion. This has been shown to be an inefficient cycle 
(Art. 417), and it would be impossible, for mechanical reasons, to 
more than approximate it in practice. The inlet port is nearly 

always closed prior 
to the end of the 
stroke, producing 
such a diagram as 
debgq, Fig. 184, in 
which the supply of 
steam to the cylin- 
der is less than the 
whole volume of the 
piston displacement, 
and the work area 
under bg is obtained 
without the supply 
of heat, but solely 
in consequence of the expansive action of the steam. Apparently, 
then, the actual steam engine cycle is that of Rankine* (Art. 411). 

* It need scarcely be said that the association of the steam engine indicator dia- 
gram and its varying quantity of steam with the ideal Rankine cycle is open to objec- 
tion (Art 454). Yet there are advantages on the ground of simplicity in this method 
of approaching the subject. 




Fig. 187. Arts. 424, 425, 427, 430, 431, 436, 441, 445, 446, 448 
449, 450, 451, 452, 454. — Indicator Diagram and Rankine Cycle. 



WIREDRAWING 259 

But if we apply an indicator (Art. 484) to the cylinder, — an instru- 
ment for graphically recording the changes of pressure and volume 
during the stroke of the piston, — we obtain some such diagram as 
abodes, Fig. 187, which may be instructively compared with the cor- 
responding Rankine cycle, ABODE. The remaining study of the 
steam engine deals principally with the reasons for the differences 
between these two cycles. 

425. Wiredrawing. The first difference to be considered is that along the 
lines ab, AB. An important reason for the difference in volumes at b and B will 
be discussed (Art. 430) ; we may at present note that the jwessures at a and b are 
less than those at A and B, and that the pressure at b is less than that at a. This 
is due to the frictional resistance of steam pipes, valves, and ports, which causes 
the steam to enter the cylinder at a pressure somewhat less than that in the boiler ; 
and produces a further drop of pressure while the steam enters. The action of 
the steam in thus expanding with considerable velocity through constricted pas- 
sages is described as " wiredrawing." The average pressure along ab will not 
exceed 0.9 of the boiler pressure; it may be much less than this. A loss of work 
area ensues. The greater part of the loss of pressure occurs in the ports and pas- 
sages of the cylinder and steam chest. The friction of a suitably designed steam 
pipe is small. The pressure drop due to wiredrawing or " throttling," as it is 
sometimes called, is greatly aggravated when the steam is initially wet; Clark 
found that it might be even tripled. Wet steam may be produced as a result of 
priming or frothing in the boiler, or of condensation in the steam pipes. Its evil 
effect in this as in other respects is to be prevented by the use of a steam separator 
near the engine ; this automatically separates the steam and entrained moisture, 
and the water is then trapped away. 

The mean piston velocity in the average steam engine is about 10 ft. per sec- 
ond. A high speed of the piston as compared with the velocity of the steam might 
therefore be expected to accentuate the pressure drop along ab. The speed of the 
piston is, however, always low as compared with that of the steam, and there is 
consequently a perceptible impact when steam is admitted. This leads to a rise 
of pressure, and the shape of the line ab, so far as piston speed may affect it, is 
determined by the joint effect of these two causes. 

426. Thermodynamics of Throttling. Wiredrawing is a non-reversible 
process, in that expansion proceeds, not against a sensibly equivalent 
external pressure, but against a lower and comparatively non-resistant 
pressure. If the operation be conducted with sufficient rapidity, and if 
the resisting pressure be negligible, the external work done should be 
zero, and the initial heat contents should be equal to the final heat 
contents ; i.e. the steam expands adiabatically (though not isentropically) 
along a line of constant total heat like mr, Fig. 161. The steam is thus 
dried by throttling ; but since the temperature has been reduced, the heat 



260 



APPLIED THERMODYNAMICS 





w////////,- ~ 



^//////////////A. V////////////// G l 1 



s 

Fig. 188. Art. 426. — Throttling 
of Superheated Steaui. 



has lost availability. Figure 188 represents the case in which the steam 
remains superheated throughout the throttling process. A is the initial 

state, DA and EC lines of constant pressure, 
AB an adiabatic, AF a line of constant total 
heat, and C the final state. The areas 
SHJDAG and SHECK, and, consequently, 
the areas JDABEH and GBCK, are equal ; 
the temperature at C is less than that at A. 
(See the superheated steam tables : at p = 140, 
H= 1298.2 when £ = 553.1° P.; atp = 100, 
#=1298.2 when t is about 548° P.) The 
effect of wiredrawing is thus generally to 
lower the temperature, while leaving the total quantity of heat unchanged. 
The curve ab, Pig. 187, must, in theory at least, appear on the entropy 
diagram as a line of constant 
total heat. On the ideal entropy 
diagram ABODE of Pig. 189 we 
therefore sketch the "wire- 
drawn " line ab. 

427. Regulation by Throttling. 
On some of the cheaper types of 
steam engine, the speed is controlled 
by varying the extent of opening of 
the admission pipe, thus producing 
a wiredrawing effect throughout the 
stroke. It is obvious that -such a 
method of regulation cannot be 
other than wasteful ; a better method is, as in good practice, to vary the point of 
cut-off, b, Fig. 187. 

428. Expansion Curve. The widest divergence between the theoretical 
and actual diagrams appears along the expansion lines be, BC, Pig. 187. 
In neither shape nor position do the two lines coincide. Early progress in 
the development of the steam engine resulted in the separation of the 
three elements, boiler, cylinder, and condenser. In spite of this separa- 
tion, the cylinder remains, to a certain extent, a condenser as well as a 
boiler, alternately condensing and evaporating large proportions of the 
steam supplied, and producing erratic effects not only along the expansion 
line, but at other portions of the diagram as well. 




Fig. 



189. Arts. 426, 445, 453. — Converted Indi- 
cator Diagram and Rankine Cycle. 



429. Importance of Cylinder Condensation. The theoretical analysis of the Ran- 
kine cycle (Art. 411) gives efficiencies considerably greater than those actually attained 
in practice. The reason for this was not suggested by Rankine, who in his earlier 



CYLINDER CONDENSATION 261 

writings almost wholly ignored the fact that nothing approaching an adiabatic 
condition is possible with steam contained in the conducting cast-iron walls of an 
engine cylinder. The actual action was pointed out by Clark's experiments on 
locomotives in 1855 (1) ; and still more comprehensively by Isherwood, in his 
classic series of engine trials made on a vessel of the United States Navy (2). 
The further studies of Loring and Emery and of Ledoux (3), and, most of all, 
those conducted under the direction of Him (4), served to point out the vital 
importance of the question of heat transfers within the cylinder. Recent accurate 
measurements of the fluctuations in temperature of the cylinder walls by Hall, 
Callendar and Nicolson (5) and at the Massachusetts Institute of Technology (6) 
have furnished quantitative data. 

430. Initial Condensation. When hot steam enters the cylinder at or 
near the beginning of the stroke, it meets the relatively cold surface of 
the piston and cylinder head, and partial liquefaction immediately occurs. 
As the piston moves forward, more of the cylinder wall is exposed to the 
steam, and condensation continues. The initial condensation is by far the 
most important of the heat exchanges to be considered. By the time 
the point of cut-off is reached the steam may contain from 25 to 70 per 
cent of water. The actual weight of steam supplied by the boiler is, 
therefore, not determined by the volume at b, Fig. 187 ; it is practically 
from 33 to 233 per cent greater than the amount thus determined. If 
ABODE, Fig. 187, represents the ideal cycle, then b will be found at a point 
where V b = from 0.30 V B to 0.75 V B (Art. 436). 

431. Condensation during Expansion. The admission valve closes at b, 
and the steam is permitted to expand. Condensation continues for a 
time, the chilling wall surface increasing. As expansion proceeds the 
pressure of the steam falls until its temperature becomes less than that of 
the cylinder walls, when an opposite transfer of heat begins. The ivalls 
now give up heat to the steam, drying it, i.e. evaporating a portion of the 
commingled water. At the moment when the direction of heat trans- 
fer changes, the percentage of water has usually reached a maximum ; 
from that point onward, it decreases. The behavior is complicated, how- 
ever, by the liquefaction which necessarily accompanies expansion, even if 
adiabatic (Art. 372). The reevaporation of the water during the later stages 
of expansion is effected by a withdrawal of heat from the walls ; these 
are consequently cooled, resulting in the resumption of proper conditions 
for a repetition of the whole destructive process during the next succeed- 
ing stroke. Reevaporation is an absorption of heat by the fluid. For 
maximum efficiency, all heat should be absorbed at maximum temperature, 
as in the Carnot cycle. The later in the stroke that reevaporation occurs, 
the lower is the temperature of reabsorption of this heat, and the greater 
is the loss of efficiency. Reevaporation is often not thermally complete : 



262 APPLIED THERMODYNAMICS 

the steam may in some cases be brought to a condition of dryness at the 
point of release, but in general the temperature of the steam at that point 
is at least 20° F. below that of the cylinder walls (7). 

432. Continuity of Action. When unity of weight of steam condenses, it gives 
up the latent heat L ; when afterward reevaporated, it reabsorbs the latent heat L x ; 
meanwhile, it has cooled, losing the heat h — h r The net result is an increase of 
heat in the walls of L — L x + h — h x — H — H v and the walls would continually be- 
come hotter, were it not for the fact that heat is being lost by radiation to the 
external atmosphere and that more water is reevaporated than was initially con- 
densed ; so much more in fact, that the dryness at the end of expansion is usually 
greater than it woidd have been, had expansion been adiabatic. 

The outer portion of the cylinder walls remains at practically uniform tem- 
perature, steadily and irreversibly losing heat to the atmosphere. The inner portion 
has been experimentally shown to fluctuate in temperature in accordance with the 
changes of temperature of the steam in contact with it. The depth of this "peri- 
odic " portion is small, and decreases as the time of contact during the cycle 
decreases, e.g. in high speed engines. 

433. Influences Affecting Condensation. Four main factors are related 
to the phenomena of cylinder condensation : they are (a) the temperature 
range, (b) the size of the engine, (c) its speed and (most important), 
(d) the ratio of volumes during expansion. Of extreme importance, as 
affecting condensation during expansion, is the condition of the steam at the 
beginning of expansion. If this is wet, either because of the delivery of 
wet steam to the engine or because of initial condensation (Art. 430), the 
condensation during expansion is greatly increased. 

The greater the range of pressures (and temperatures) in the engine, the more 
marked are the alternations in temperature of the walls, and the greater is the 
difference in temperature between steam and walls at the moment when steam is 
admitted to the cylinder. A wide range of working temperatures, although practi- 
cally as well as theoretically desirable, has thus the disadvantage of lending itself 
to excessive losses. 

434. Speed. At infinite speed, there would be no time for the transfer of 
heat, however great the difference of temperature. Willans has shown the per- 
centage of water present at cut-off to decrease from 20.2 to 5.0 as the speed in- 
creased from 122 to 401 r. p. m., the steam consumption per Ihp-hr. concurrently 
decreasing from 27.0 to 24.2 lb. (8). In another test by Willans, the speed ranged 
from 131 to 405 r. p. m., the moisture at cut-off from 29.7 to 11.7, and the steam 
consumption from 23.7 to 20.3 ; and in still another, the three sets of figures were 
116 to 401, 20.9 to 8.9, and 20.0 to 17.3. In all cases, for the type of engine under 
consideration, increase of speed decreased the proportion of moisture and increased 
the economy : but it should not be inferred from this that high speeds are neces- 
sarily or generally associated with highest efficiency. 



EXPANSION AND CYLINDER CONDENSATION 263 

435. Size. The volume of a cylinder is ttD^L -4- 4 and its exposed wall sur- 
face is (ttDL) + (7i\D 2 -4- 2), if D denotes the diameter and L the exposed length. 
The volume increases more rapidly than the wall surface, as the , diameter is in- 
creased for a constant length. Since the lengths of cylinders never exceed a certain 
limit, it may be said, generally, that small engines show greater amounts of con- 
densation, and lower efficiencies, than large engines. 

486. Ratio of Expansion. This may be denned as V d h- V b , Fig. 187 (Art. 450). 
The greater the ratio of expansion, the greater is the liquefaction accompanying- 
expansion. This would be true even if expansion were adiabatic ; with early cut- 
off, moreover, the time during which the metal is exposed to high temperature 
steam is reduced, and its mean temperature is consequently less. Its activity as 
an agent for cooling the steam during expansion is thus increased. Again, the 
volume of steam during admission is more reduced by early cut-off than is the ex- 
posed cooling surface, since the latter includes the two constant quantities, the 
surfaces of the piston and of the cylinder head (clearance ignored (Art. 450)). 
Initial condensation is thus greatly increased when the ratio of expansion is in- 
creased, as shown by Isherwood : and, as has been shown (Art. 433), excessive 
initial condensation leads to excessive condensation during expansion. The 
following shows the results of several experiments : 



Observers 


Eatio of 

Expansion 


Per Cent, of Water 

at Cut-off 


Steam Consumption, 
Pounds per Ihp-hr. 


Loring and Emery 
Willans (9) 


Low 

4.2 
4.0 


Higli 

16.8 

8.0 


Low 

8.9 


High 
25.0 


Low 

21.2 

20.7 


High 

25.1 
23.1 



Barrus (10) gives the following as average results from a large number of tests 
of Corliss engines at normal speed : 



Cut-off, Per Cent. 


Percentage of 


Cut-off, Per Cent. 


Percentage of 


of Stroke 


Condensation 


of Stroke 


Condensation 


2.5 


62 


25.0 


24 


5.0 


54 


30.0 


20 


10.0 


44 


40.0 


16 


15.0 


36 


45.0 


15 


20.0 


28 







In these three sets of experiments, it was found that the propor- 
tion of water steadily decreased as the ratio of expansion decreased. 
The steam consumption, however, decreased to a certain minimum fig- 
ure, and then increased. The beneficial effect of a decrease in con- 
densation was here, as in general practice, offset at a certain stage 



264 APPLIED THERMODYNAMICS 

by the thermodynamic loss due to relatively incomplete expansion, 
discussed in Art. 418. The proper balancing of these two factors, 
to secure best efficiency, is the problem of the engine designer. It 
must be solved by recourse to theory, experiment, and the study of 
standard practice. In American stationary engines, the ratio of ex- 
pansion in simple cylinders is usually from 4 to 5. 

437. Quantitative Effect. Empirical formulas for cylinder condensation have 
been presented by Marks and Heck, among others. Marks (11) gives a curve 
of condensation, showing the proportion of steam condensed for various ratios of 
expansion, all other factors being eliminated. A more satisfactory relation is 
established by Heck (12), whose formula is 

f/N\pe 

in which M is the proportion of steam condensed at cut-off, N is the speed of the 
engine (r. p. m.), s is the quotient of the exposed surface of the cylinder in 
square feet by its volume in cubic feet, p is the absolute pressure per square inch 
at cut-off, e is the reciprocal of the ratio of expansion, and T is the temperature at 
cut-off. Heck estimates that the steam consumption of an engine may be com- 
puted from its indicator diagram (Art. 500) within 10 per cent, by the application 
of this formula. If the steam as delivered from the boiler is wet, some modifica- 
tion is necessary. 

438. Reduction of Condensation. Aside from careful attention to 
the factors already mentioned, the principal methods of minimizing 
cylinder condensation are by (a) the use of steam jackets, (5) super- 
heating the steam, and (c) the employment of multiple expansion. 

439. The Steam Jacket. The thermal interchange represented by the 
expression L — L x + h — h Y of Art. 432 involves a continuous supply of 
heat to the cylinder walls, which may be expressed from Art. 360 as 
0.305 (t — t{). This heat is removed from the walls in one of three ways: 
because of (a) water entering the cylinder with the steam, (b) liquefac- 
tion accompanying expansion (the excess of moisture actively abstracting 
heat (Art. 431)), or (c) the transfer of heat from the cylinder walls to the 
atmosphere. To maintain thermal equilibrium, the steam must supply to 
the walls sufficient heat to offset these losses. If we can reduce any one 
of the latter, then the expenditure of heat by the steam will be correspond- 
ingly reduced. The simple expedient of covering or " lagging " the barrel 
and head of the cylinder is intended to reduce initial condensation by de- 
creasing the loss of heat to the atmosphere. 

The steam jacket, invented by Watt, is a hollow casing enclosing the 
cylinder walls, within which steam is kept at high pressure. Jackets 



STEAM JACKETS 265 

have often been mechanically imperfect, and particular difficulty has been 
experienced in keeping then drained of the condensed water. In a few 
cases, the steam has passed through the jacket on its way to the cylinder ; 
a bad arrangement, as the cylinder steam was thus made wet. It is usual 
practice, with simple engines, to admit steam to the jacket at full boiler 
pressure ; and in some cases the pressure and temperature in the jacket 
have exceeded those in the cylinder. Hot-air jackets have been used, in 
which flue gas from the boiler, or highly heated air, was passed about the 
body of the cylinder. 

440. Arguments for and against Jackets. The exposed heated surface 
of the cylinder is increased and its mean temperature is raised; the 
amount of heat lost to the atmosphere is thus increased. The jacket is at 
one serious disadvantage: its heat must be transmitted through the entire 
thickness of the walls ; while the internal heat transfers are effected by 
direct contact between the steam and the inner " skin " of the walls. The 
use of a jacket might seem likely to lead to excessive heating of the steam 
during the exhaust stroke, thus raising the pressure and causing a resistance 
to the movement of the piston. The fact is, however, that no such effect is 
produced, because the steam is dry or nearly so, and practically a non-con- 
ductor of heat, during the exhaust stroke. Un jacketed cylinder walls act 
like heat sponges. The difference in mean temperature between walls and 
steam would not alone account for excessive condensation, if the steam 
initially were dry. Small proportions of moisture greatly facilitate the 
heat transfers. 

The function of the jacket is preventive, rather than remedial, oppos- 
ing the formation of moisture early in the stroke, liquefaction being 
transferred from the cylinder to the jacket, where its influence is less 
harmful. The walls are kept hot at all times, instead of being periodi- 
cally heated and cooled by the action of the cylinder steam. The steam 
in the jacket does not expand ; its temperature is at all times the maxi- 
mum temperature attained in the cycle. The mean temperature of the 
walls is thus raised ; it may even be equal to that of the steam during ad- 
mission, instead of being 50° lower, as was found by Donkin, with an un- 
jacketed cylinder. The detrimental influence of the walls is in all cases 
mitigated ; the working fluid in the cylinder is, on the whole, gaining 
rather than losing heat during expansion. The higher mean temperature 
of the walls makes reevaporation begin earlier, and thus raises the tem- 
perature of reception of the proportion of total heat thus supplied. 

441. Results of Jacketing. In the ideal case, the action of the jacket may be 
regarded as shown by the difference of the areas dekl and debf, Fig. 183. The 
total heat supplied, without the jacket, is ldeb2, but cylinder condensation makes 



266 



APPLIED THERMODYNAMICS 




'12 V 8 "/ 16 "/ 16 

POINT OF CUT-OFF 



the steam wet at cut-off, giving the work area dekl only. The additional heat 
26/3, supplied by the jacket, gives the additional work area kbfl, manifestly at 
high efficiency. In this country, jackets have been generally employed on well- 
known engines of high efficiency, particularly on slow speed pumping engines ; but 
their use is not common with standard designs. Slow speed and extreme expan- 
sion, which suggest jackets, lead to excessive bulk and first cost of the engine. 
With normal speeds and expansive ratios, the engine is cheaper and the necessity 
for the jacket is less. The use of the jacket is to be determined from considera- 
tions of capital charge, cost of fuel and load factor, as well as of thermodynamic 
efficiency. These commercial factors account for the far more general use of the 
jacket in Europe than in the United States. 

From 7 to 12 per cent of the whole amount of steam supplied to the engine 
may be condensed in the jacket. The power of the engine is invariably in- 
creased by a greater percentage than that of increase 
of steam consumption. The cylinder saves more than 
the jacket spends, although iu some cases the amount 
of steam saved has been small. The range of saving 
may be from 2 or 3 up to 25 per cent. The in- 
creased power of the engine is represented by the 
difference between the areas abcdes and aXYCes, 
Fig. 187. The latter area approaches much more 
closely the ideal area ABCDE. Jacketing pays 
best when the conditions are such as to naturally 
induce excessive initial condensation. The diagram 
of Fig. 190, after Donkin (14), shows the variation 
in value of a steam jacket at varying ratios of expansion in the same engine run 
at constant speed and initial pressure. 

442. Use of Superheated Steam. The thermodynamic advantage of 
superheating, though small, is not to be ignored, some heat being taken 
in at a temperature higher than the mean temperature of heat absorption ; 
the practical advantages are more important. By superheating, a smaller 
weight of steam may be made to deliver a given quantity of heat to the 
cylinder. Adequate superheat fills the " heat sponge " formed by the 
walls, without letting the steam become wet in consequence. If super- 
heating is slight, the steam, during admission, may be brought down to 
the saturated condition, and may even become wet at cut-off, following 
such a path as debxbkl, Fig. 183. With a greater amount of superheat, 
the steam may remain dry or even superheated at cut-off, giving the paths 
debzyf, debkzA. The minimum amount of superheat ordinarily necessary 
to give dryness at cut-off seems to be about 100° F. ; it may be much 
greater. Kipper finds (15) that about 7.5° F. of superheat are necessary 
for each 1 per cent of wetness at cut-off to be expected when working 
with saturated steam. We thus obtain Fig. 191, in which the increased 
work areas acbd, cefb, eghf&ve obtained by superheating along jk, M, Im, 
each path representing 75° of superheat. Taking the pressure along ag 



Fig. 190. Art. 441. — Effect 
of Jackets at Various Ex- 
pansion Ratios. 



SUPERHEAT 



267 



as 120 lb., and that along lib as 1 lb., the absolute temperatTires are 800.9° 
and 561.43°, respectively, and since the latent heat at 120 lb. is 877.2 
B. t. u., the work gained by each of the areas in 
question is 

'800.9 - 561.43 N 



87.72 



800.9 



= 26.1 B. t. u. 



my 
ac e(J jy 


I o o o o \ 

/ o ooo \ 

/ 888K y 

/ \ 

/ dhfh \ 









N 

Fig. 191. Art. 442. — Super- 
heat for overcoming Initial 
Condensation. 




N 



If we take the specific heat of superheated 
steam, roughly, at 0.48, the heat used in secur- 
ing this additional work area is 0.48 x 75 = 36 
B. t. u. The efficiency of superheating is then 
26.1 -f- 36 = 0.73, while that of the non-super- 
heated cycle as a whole, even if operated at Car- 
not efficiency, cannot exceed 239. 47 -=-800.9 = 0.30. 
Great care should be taken to avoid loss of heat in pipes between the super- 
c heater and the cylinder; without thorough 
insulation the fall of temperature here may 
be so great as to considerably increase the 
amount of superheating necessary to secure 
the desired result in the cylinder. 

The actual path due to superheating in 
practice is not quite as simple as those sug- 
gested in Fig. 183. In Fig. 192, let the path 
as heretofore conceived be ABCFG. If there 
is wiredrawing during admission, the pres- 
sure at cut-off may be represented by the 
line HJ, and the path CFQ will be replaced 
by CM, KL being a line of constant total heat through F. Expansion 
then begins at M instead of F. 

443. Experimental Results with Super- 
heat. The Alsace tests of 1892 showed, with 
from 60° to 80° of superheat, an average net 
saving of 12 per cent, even when the coal con- 
sumed in the separately fired superheaters 
was considered ; and when the superheaters 
were fired by waste heat from the boilers, 
the average saving was 20 per cent. Willans 
found a considerable saving by superheat, 
even when cut-off was at half stroke, a ratio 
of expansion certainly not unduly favorable 
to superheating. As with jackets, the ad- 
vantage of superheat is greatest in engines 
of low speeds and high expansive ratios. Striking results have been obtained by 
the use of high superheats, ranging from 200° to 300° F. above the temperature 



Fig. 192. Art. 442. — Superheat 
as affected by Radiation and 
Wiredrawing. 



650- 

=>600- 
o 

x 550- 




MfJ' 


S 5 oo- 

S450- 
< 




£400- 






Z 350 " 






°300- 












£250- 






w 200- 






-j 150- 






< 






£ioo- 






*- 50 








5 


10 15 20 



INDICATED HORSE POWER 

Fig. 193. Art. 443, Prob. 7. — Steam 
Economy in Relation to Superheat. 



268 



APPLIED THERMODYNAMICS 



of saturation. The mechanical design of the engine must then be considerably 
modified. Vaughan (16) has reported remarkably large savings due to superheat- 
ing in locomotive practice. Figure 193 shows the decreased steam consumption 
due to various degrees of superheat in a small high-speed engine. 



444. Superheat vs. High Pressure. In Fig. 194, the work area CEFHD is 
gained as a result of superheating along EF. This 
may even exceed the additional heat absorbed, JEFK, 
on account of the reduction of initial condensation. 
By increasing the initial pressure, the area BLNC 
might have been gained, but at an expenditure for heat 
of (practically) LMEB, always greater than the addi- 
tional work obtained. With efficient superheaters, a 
given amount of heat in superheated steam may be 
delivered to the engine at the same cost as the same 
amount of heat in saturated steam ; but the latter 
gives a less efficient cycle in the cylinder than the 
former. High pressure soon reaches a mechanical 
limit; the limit is not as quickly reached with superheat, although minor diffi- 
culties in lubrication have been experienced. 



J N 




F 

A 




• 


c 




/ 


D 


\ 


H 


/ 




j 


\ 

k 



Fig. 194. Art. 444. — Super 
heat vs. High Pressure. 



445. Actual Expansion Curve. In Fig. 187, bY represents the 
curve of saturation, bO the adiabatic. The actual expansion curve 
in an unjacketed cylinder using saturated steam will then be some 
such line as be, the entropy and fraction of dryness xy -*- xz first de- 
creasing (condensation) and afterwards increasing (reevaporation) 
as expansion proceeds. Expressed exponentially, the value of n for 
such expansion curve is less than that for the adiabatic or the curve 
of saturation; in actual practice it is always close to 1.00, whence 
the equation of the curve is P V— pv. It should not be confused with 
the perfect gas isothermal ; that the equation has the same form is 
accidental. The curve PV=pv is an equilateral hyperbola, com- 
monly called the hyperbolic line. It may be plotted for comparison 
with expansion lines of actual indicator diagrams by the methods of 
Arts. 92, 93. 

The actual expansion line be of Fig. 187 then appears as bze, 
Fig. 189. Heat is first lost to the walls ; the expansion line then 
recrosses the adiabatic (note the point Jf, Fig. 187), while re- 
evaporation causes heat absorption along zc. The heat given up 
to the walls is bzmn ; that reabsorbed equals zeom. 



MEAN EFFECTIVE PRESSURE 269 

446. Work done during Expansion : Engine Capacity. From 
Art. 95, this is, for a hyperbolic curve, Fig 187, P b V b log e —S • As- 

sume admission and exhaust to occur without change of pressure ; 
the cycle is then precisely that represented by ABCDU, excepting 
that the expansive path is hyperbolic. Then the work done during 
admission is P B V B \ the negative work during exhaust is P D V^\ 
and the net work of the cycle is 

p b v b + p b v B io ge |? - p B v c = p B r B (i + io ge -g - p D v c . 

The mean effective pressure or average ordinate of the work area is 
obtained by dividing this by V c , giving 

Vr 



PbV- b [1 + \o 



or, letting — 9- = r, it is 
Vb 



V c 



P*(l + lo£ e r) 



Ill^P, 



-Pi 



Letting m stand for this mean effective pressure, in pounds per 
square inch, A for the piston area in square inches, L for the length 
of the stroke in feet, and iVfor the revolutions per minute, the total 
average pressure on the piston is mA pounds, the distance which it 
moves per minute is 2 LN feet, and for a double-acting engine the 
work per minute is 2 mALN foot-pounds, or 2 mALN-i- 33,000 horse 
power. This is for an ideal diagram, which is always larger than the 
actual diagram abcdes ; the ratio of the latter to the former gives the 
diagram factor, by which the computed value of m must be multiplied 
to give actual results. 

Diagram factors for various types of engine, as given by Seaton, are as 
follows : — 

Expansion engine, with special valve gear, or with a separate cut-off valve, 
cylinder jacketed . . .0.90; 

Expansion engine having large ports and good ordinary valves, cylinders 
jacketed . . . 0.86 to 0.88 ; 

Expansion engines with ordinary valves and gear as in general practice, and 
unjacketed . . . 0.77 to 0.81 ; 



270 APPLIED THERMODYNAMICS 

Compound engines, with expansion valve on high pressure cylinder, cylinders 
jacketed, with large ports, etc. . . . 0.86 to 0.88 ; 

Compound engines with ordinary slide valves, cylinders jacketed, good ports, 
etc. . . . 0.77 to 0.81 ; 

Compound engines with early cut-off in both cylinders, without jackets or 
separate expansion valves . . . 0.67 to 0.77 ; 

Fast-running engines of the type and design usually fitted in warships 
. . . 0.57 to 0.77. 

447. Capacity vs. Economy. If we ignore the influence of con- 
densation, the Clausius cycle (Art. 409), objectionable as it is with 
regard to capacity (Art. 418% would be the cycle of maximum effi- 
ciency ; practically, when we contemplate the excessive condensation 
that would accompany anything like complete expansion, the cycle of 
Rankine is superior. This statement does not apply to the steam tur- 
bine (Chapter XIV). The steam engine may be given an enormous 
range of capacity by varying the ratio of expansion ; but when this 
falls above or below the proper limits, economy is seriously sacrificed. 
In purchasing engines, the ratio of expansion at normal load should 
be set fairly high, else the overload capacity will be reduced. In 
marine service, economy of fuel is of especial importance, in order to 
save storage space. Here expansive ratios may therefore range 
higher than is common in stationary practice, where economy in first 
cost is a relatively more important factor. 

448. The Exhaust Line : Back Pressure. Considering now the line de of Fig. 
187, we find a noticeable loss of work area as compared with that in the ideal 
case. (Line BE represents the pressure existing outside the cylinder.) This is 
due to several causes. The frictional resistance of the ports and exhaust pipes 
(greatly increased by the presence of water) produces a wiredrawing effect, mak- 
ing the pressure in the cylinder higher than that of the atmosphere or of the con- 
denser. The presence of air in the exhaust passages of a condensing engine may 
elevate the pressure above that corresponding to the temperature of the steam, 
and so cause undesirable resistance to the backward movement of the piston. 
This air may be present as the result of leakage, under poor operating conditions ; 
more or less air is always brought in the cycle with the boiler feed and condenser 
water. The effect of these causes is to increase the pressure during release, even 
in good engines, from 1.3 to 3.3 lb. above that ideally obtainable. 

Reevaporation may be incomplete at the end of expansion ; it then proceeds 
during exhaust, sometimes, in flagrant cases, being still incomplete at the end of 
exhaust. The moisture then present greatly increases initial condensation. The 
evaporation of any water during the exhaust stroke seriously cools the cylinder 
walls ; but it also increases the pressure resisting the movement of the piston and 



CLEARANCE AND COMPRESSION 



271 



thus raises the mean elevation of the line de, Fig. 187. In general good practice, 
the steam is about dry during exhaust ; or at least during the latter portion of the 
exhaust. 

449. Effect of Altitude. The possible capacity of a non-condensing engine is 
obviously increased at low barometric pressures, on account of the lowering of the 
line DE, Fig. 187. With condensing engines, the absolute pressure attained along 
DE depends upon the proportion of cooling water supplied and the effectiveness 
of the condensing apparatus. It is practically independent of the barometric pres- 
sure, excepting at very high vacua ; consequently, the capacity of the engine is 
unchanged by variations in the latter. A slightly decreased amount of power, 
however, will suffice to drive the air pump which delivers the products of conden- 
sation against any lessened atmospheric pressure. 



450. Clearance. The line esa does not at any point come in contact with the 
ideal line EA, Fig. 187. In all engines, there is necessarily a small space left 
between the piston and the inside of the cylinder head at the end of the stroke. 
This space, with the port spaces back to the contact surfaces of the inlet valves, is 
filled with steam throughout the cycle. The distance ts in the diagram represents the 
volume of these " clearance " spaces. The 
expansion line be is hyperbolic with ref- 
erence to the axis OP ; and by a simple re- 
versal of Art. 92 and Art. 93, the approxi- 
mate location of this axis may readily be 
found from any actual diagram. In Fig. 
195, the apparent ratio of expansion is 

fD 

— • If the zero volume line OP be found, 
ab 

the real ratio of expansion, clearance vol- 

FD 

ume included, is —ry The clearance in 
Ab 

actual engines varies from 2 to 10 per 
cent of the piston displacement, the nec- 
essary amount depending largely on the type of valve gear. In such an engine 
as that of Fig. 186, it is necessarily large, on account of the long ports. It is 
proportionately greater in small engines than in those of large size. It may be 
accurately estimated by placing the piston at the end of the stroke and filling the 
clearance spaces with a weighed or measured amount of water. 



p 

a 


h 




f Y D| 




h (1 






Fig. 195. Arts. 450, 451. — Real and Ap- 
parent Expansion. 



451. Compression. A large amount of steam is employed to fill the clearance 
space at the beginning of each stroke. This can be avoided by closing the exhaust 
valve prior to the end of the stroke, as at e, Fig. 187, the piston then compressing 
the clearance steam along es, so that the pressure is raised nearly or quite to that 
of the entering steam. This compression serves to bring the piston gently to rest, 
without shock, at the end of the exhaust stroke. If compression is so complete as 
to raise the pressure of the clearance steam to that carried in the supply pipe, no 
loss of steam will be experienced in filling clearance spaces. The work expended 



272 APPLIED THERMODYNAMICS 

in compression eahg, Fig. 195, will be largely recovered during the next forward 
stroke by the expansion of the clearance steam : the clearance will thus have had 
little effect on the efficiency ; the loss of capacity efa will be just balanced by the 
saving of steam, for the amount of steam necessary to fill the clearance space 
would have expanded along ae, if no other steam had been present. 

Complete compression would, however, raise the temperature of the com- 
pressed steam so much above that of the cylinder walls that serious condensation 
would occur. This might be counteracted by jacketing, but in practice it is cus- 
tomary to terminate compression at some pressure lower than that of the entering 
steam. A certain amount of unresisted expansion then takes place during the 
entrance of the steam, giving a wiredrawn admission line. If the pressure at s, 
Fig. 187, is fixed, it is, of course, easy to determine the point e at which the 
exhaust valve must close. Considered as a method of warming the cylinder walls 
so as to prevent initial condensation, compression is " theoretically less desirable 
than jacketing, for in the former case the heat of the steam, once transformed to 
work, with accompanying heavy losses, is again transformed into heat, while in 
the latter, heat is directly applied." For mechanical reasons, some compression is 
usually considered necessary. It makes the engine smooth-running and probably 
decreases condensation if properly limited. Compression must not be regarded as 
bringing about any nearer approach to the Carnot cycle. It is applied to a very 
small portion only of the working substance, the major portion of which is 
externally heated during its passage through the steam plant. 

452. Valve Action. We have now considered most of the differences between 
the actual and ideal diagrams of Fig. 187. The rounding of the corners at b, and 
along cdu, is due to sluggish valve action ; valves must be opened slightly before the 
full effect of their opening is realized, and they cannot close instantaneously. The 
round corner at e is due to the slow closing of the exhaust valve. The inclined line 
sa shows the admission of steam, the shaded work area being lost by the slow move- 
ment of the valve. In most cases, admission is made to occur slightly prior to the 
end of the stroke, in order to avoid this very effect. If admission is too early, a 
negative lost work loop, mno, may be formed. Important aberrations in the diagram, 
and modifications of the phenomena of cylinder condensation, may result from 
leakage past valves or pistons : these are matters of operating error, beyond the 
scope of the present study. 

The Steam Engine Cycle on the Entropy Diagram 

453. Cylinder Feed and Cushion Steam. Fig. 189 has been left incomplete, for 
reasons which are now to be considered. It is convenient to regard the working 
fluid in the cylinder as made up of two masses, — the "cushion steam," which 
alone fills the compression space at the end of each stroke, and is constantly present, 
and the " cylinder feed," which enters at the beginning of each stroke, and leaves 
before the completion of the next succeeding stroke. In testing steam engines by 
weighing the discharged and condensed steam, the cylinder feed is alone measured ; 
it alone is chargeable as heat consumption ; but for an accurate conception of the 
cyclical relations in the cylinder, the influence of the cushion steam must be con- 
sidered. 



CONVERTED DIAGRAMS 



273 



In Fig. 196, let abcde be the PV diagram of the mixture of cushion steam and 
cylinder feed, and let gh be the expansion line of the cushion steam if it alone were 
present. The total volume vq at any point q of the combined paths is made up 
of the cushion steam volume vo and the 
cylinder feed volume, obviously equal to 
oq. If we wish to obtain a diagram 
showing the behavior of the cylinder 
feed alone, we must then deduct from 
the volumes around abcde the correspond- 
ing volumes of cushion steam. The point 
p is then derived by making vp = vq — vo, 
and the point t by making rt = ru — vs. 
Proceeding thus, we obtain the diagram r 
nzjklm, representing the behavior of the 
cylinder feed. Along nz the diagram r. 
coincides with the OP axis, indicating o 
that at this stage the cylinder contains p IG -^ 
cushion steam only. 



p 


i K b 

\ \ 
\ \ 
\ \ 
\ \ 
\ \ 

\ -\ \ 




\ '' l r 
V /» \ d 





Arts. 453, 457. — Elimination of 
Cushion Steam. 



454. The Indicator Diagram. Our study of the ideal cycles in Chapter XII has 
dealt with representations on a single diagram of changes occurring in a given mass 
of steam at the boiler, cylinder, and condenser, the locality of changes of condition 
being ignored. The energy diagram abcdes of Fig. 187 does not represent the 
behavior of a definite quantity of steam working in a closed cycle. The pressure 
and volume changes of a varying quantity of fluid are depicted. During expansion, 
along be, the quantity remains constant ; during compression along es, the quantity 
is likewise constant, but different. Along sab the quantity increases ; while along 
cde it decreases. The quality or dryness of the steam along es or be may be readily 
determined by comparing the actual volume with the volume of the same weight 
of dry steam ; but no accurate information as to quality can be obtained along the 
admission and release lines sab and cde. The areas under these lines represent 
work quantities, however, and it is therefore possible to draw an entropy diagram 
which shall represent the corresponding heat expenditures. Such a diagram will 

not give the thermal history of any definite 
amount of steam ; it is a mere projection of 
the PV diagram on different coordinates. 
It tacitly assumes the indicator diagram to 
represent a reversible cycle, whereas in fact 
the operation of the steam engine is neither 
cyclic nor reversible. 

455. Boulvin's Method. In Fig. 197, 
let abcde be any actual indicator diagram, 
YZ the pressure temperature curve of 
saturated steam, and QR the curve of satu- 
Q . ration, plotted for the total quantity of 

Fig. 197. Art. 455. -Transfer from P V steam in the cylinder during expansion. 
to NT Diagram (Boulvin's Method). The water line OS and the saturation curve 



z. 




1 
D 


E 


h 


r\ 








\ 

Y 


/ 


La 


I 


\m 




L c 


B 


hX. 


G 






t\ 




\ 






V 











274 APPLIED THERMODYNAMICS 

MT are now drawn for this same quantity of steam, on the entropy plane. To 
transfer any point, like B, to the entropy diagram, we draw BD, DK, EH, KT. 
BA, AT, HT, BG, and GF as in Art. 378. Then F is the required point on the 
temperature entropy diagram. By transferring other points in the same way, we 
obtain the area NVFU, representing a reversible cycle equivalent to the actual 
diagram so far as heat quantities are concerned. The expansion line thus traced 
correctly represents the actual history of a definite quantity of fluid ; the com- 
pression line is imaginary, because during compression a much less quantity of 
fluid is actually present than that assumed. It is not safe to make deductions as 
to the condition of the substance from the NT diagram, excepting along the 
expansion curve. For example, the diagram apparently indicates that the dryness 
is decreasing along the exhaust line SU; although we have seen (Art. 448) that 
at this stage the dryness is usually increasing (17). 

456. Application in Practice. In order to thus plot the entropy diagram, it is 
necessary to have an average indicator card from the engine, and to know the 
quantity of steam in the cylinder. This last is determined by weighing the dis- 
charged condensed steam during a definite number of strokes and adding the 
quantity of clearance steam, assuming this to be just dry at the beginning of 
compression, an assumption closely substantiated by numerous experiments. 

457. Reeve's Method. By a procedure similar to that described in Art. 453, an 
indicator diagram is derived from that originally given, representing the behav- 
ior of the cylinder feed alone, on the assumption that the clearance steam works 
adiabatically through the point e, Fig. 196. This often gives an entropy diagram 
in which the compression path passes to the left of the water line, on account of 
the fact that the actual path of the cushion steam is not adiabatic, but is occa- 
sionally less " steep." 

The Reeve diagram accurately depicts the process between the points of cut- 
off and release and those of compression and admission so far as the cylinder feed 
is concerned, only. For the rest of the cycle, the entropy diagram is rather 
unsatisfactory as a method of depicting the action in the steam engine cylinder. 

458. Specimen Diagrams. In Fig. 198, 
the heat lost along ab is nearly all regained 
along be ; but it here comes back at reduced 
temperature, and consequently with reduced 
availability. Figure 199 shows the gain by 
high initial pressure and reduced back pres- 
sure. The augmented work areas befc, cfho, 
are gained at high efficiency ; adji and adlk cost 
nothing. The operation of an engine at back 
pressure, to permit of using the exhaust steam 
Fig. 198. Art. 458. — Condensation f or heating purposes, results in such losses as 
and Reevaporation. ^.. ^ Similar gaing and losseg may be 

shown for non-expansive cycles. Figure 200 shows four interesting diagrams 
plotted from actual indicator cards from a small engine operated at constant 



T 




/J 


a 

T 

N 



MULTIPLE EXPANSION 



275 



speed, initial pressure, load, and ratio of expansion (18). Diagrams A and C 
were obtained with saturated steam, B and D with superheated steam. In A and 
B the cylinder was unjacketed; in C and D it was jacketed. The beneficial in- 





Fig. 199. Art. 458. — Initial Pressure and 
Back Pressure. 



Fig. 200. Art. 458. — Effects of Jacket- 
ing and Superheating. 



fluence of the jackets is clearly shown, but not the expenditure of heat in the 
jacket. The steam consumption in the four cases was 45.6, 28.4, 27.25 and 
20.9 lb. per Ihp-hr., respectively. 



Multiple Expansion 

459. Desirability of Complete Expansion. It is proposed to show that a large 
ratio of expansion is from every standpoint desirable, excepting as it is offset by 
increased cylinder condensation ; and to suggest multiple expansion as a method 
for attaining high efficiency by making such large ratio practically possible. 

From Art. 446, it is obvious that the maximum work obtainable from a cylinder is 
a function solely of the initial pressure, the back pressure, and the ratio of expan- 
sion. In a non-conducting cylinder, maximum efficiency would be realized when 
the ratio of expansion became a maximum between the pressure limits. Without 
expansion, increase of initial pressure very slightly, if 
at all, increases the efficiency. Thus, in Fig. 201, p \ 

the cyclic work areas abed, aefg, ahij, would all be 
equal if the line XY followed the law pv = PV. 

As the actual law (locus of points representing J LA/ 

steam dry at cut-off) is approximately, 

IV ^ TT 1 7 

jpyTS = P] T6, 

the work areas increase slightly as the pressure in- 
creases ; but the necessary heat absorption also 
increases, and there is little or no net gain. The 
thermodynamic advantage of high initial pressure is 
realized only when the ratio of expansion is large. 

By condensing the steam as it flows from the engine, its pressure may be re- 
duced from that of the atmosphere to an absolute pressure possibly 13 lb. lower. 
The cyclic work area is thus increased ; and since the reduction of pressure is ac- 
companied by a reduction in temperature, the potential efficiency is increased. 
Figure 202 shows, however, that the percentage gain in efficiency is small with no 



Fig. 201. Art. 459. — Non- 
expansive Cycles. 



276 



APPLIED THERMODYNAMICS 



expansion, increasing as the expansion ratio increases. Wide ratios of expansion are 

from all of these standpoints essential to efficiency. 

We have found, however, that wide ratios of 
expansion are associated with such excessive losses 
from condensation that a compromise is necessary, 
and that in practice the .best efficiency is secured 
with a rather limited ratio. The practical attain- 
ment of large expansive ratios without correspond- 
ing losses by condensation is possible by multiple 
expansion. By allowing the steam to pass suc- 
cessively through two or more cylinders, a total 

Fig. 202. Art. 459. — Gain due expansion of 10 to 25 may be secured, with condensa- 
to Vacuum. tion losses such as are due to much lower ratios. 




460. Condensation Losses in Compound Cylinders. The range of pres- 
sures, and consequently of temperatures, in any one cylinder, is reduced 
by compounding. It may appear that the sum of the losses in the two 
cylinders would be equal to the loss in a single simple cylinder. Three 
considerations may serve to show why this is not the case : 

(a) Steam reevaporated during the exhaust stroke is rendered avail- 
able for doing work in the succeeding cylinder, whereas in a simple 
engine it merely causes a resistance to the piston. 

(b) Initial condensation is decreased because of the decreased fluctua- 
tion in wall temperature. 

(c) The range of temperature in each cylinder is half what it is in the 
simple cylinder, but the whole wall surface is not doubled. 

461. Classification. Engines are called simple, compound, triple, or quadruple, 

according to the number of successive expansion stages, ranging from one to four. 
A multiple-expansion engine may have any number of cylinders; a triple expan- 
sion engine may, for example, have five cylinders, a single high -pressure cylinder 
discharging its steam to two succeeding cylinders, and these to two more. In a 
multiple-expansion engine, the first is called high-pressure cylinder and the last 
the low-pressure cylinder. The second cylinder in a triple engine is called the 
intermediate; in a quadruple engine, the second and third are called the first 
intermediate and the second intermediate cylinders, respectively. Compound en- 
gines having the two cylinders placed end to end are described as tandem ; those 
having the cylinders side by side are cross-compound. This last is the type most 
commonly used in high-grade stationary plants in this country. The engines may 
be either horizontal or vertical : the latter is the form generally used for triples or 
quadruples, and in marine service. Sometimes some of the cylinders are horizon- 
tal and others vertical, giving what, in the two-expansion type, has been called the 
angle compound. Compounding may be effected (as usually) by using cylinders of 
various diameters and equal strokes ; or of equal diameters and varying strokes, 
or of like dimensions but unequal speeds (the cylinders driving independent 
shafts), or by a combination of these methods. 



WOOLF COMPOUND ENGINE 



277 



462. Incidental Advantages. Aside from the decreased loss through cylinder 
condensation, multiple-expansion engines have the following points of superiority : 

(1) The steam consumed in filling clearance spaces is less, because the high- 
pressure cylinder is smaller than the cylinder of the equivalent simple engine. 

(2) Compression in the high-pressure cylinder may be carried to as high a 
pressure as is desirable without beginning it so early as to greatly reduce the work 
area. 

(3) The low-pressure cylinder need be built to withstand a fraction only of 
the boiler pressure; the other cylinders, which carry higher pressures, are com- 
paratively small. 

(4) In most common types, the use of two or more cylinders permits of using 
a greater number of less powerful impulses on the piston thau is possible with a 
single cylinder, thus making the rotative speed more uniform. 

(5) For the same reason, the mechanical strains on the crank pin, shaft, etc., 
are lessened by compounding. 

These advantages, with that of superior economy of steam, have led to the 
general use of multiple expansion in spite of the higher initial cost which it en- 
tails, wherever steam pressures exceed 100 lb. 




Woolf Engine. 



463. Woolf Engine. This was a form of compound engine originated by Horn- 
blower, an unsuccessful competitor of Watt, and revived by Woolf in 1800, after 
the expiration of Watt's principal patent. 
Steam passed directly from the high to the 
low-pressure cylinder, entering the latter 
wmile being exhausted from the former. 
This necessitated having the pistons either 
in phase or a half revolution apart, and 
there was no improvement over any other 
double-acting engine with regard to uni- 
formity of impulse on the piston. Figure 

203 represents the ideal indicator diagrams, Fig. 203. Arts. 463 466. 
ABCD is the action in the high-pressure 
cylinder, the fall of pressure along CD being due to the increase in volume of 
the steam, now passing into the low-pressure cylinder and forcing its piston out- 
ward. EFGH shows the action in the low-pres- 
sure cylinder; steam is entering continuously 
throughout the stroke along EF. By laying off 
MP = LK, etc., we obtain the diagram TABRS, 
representing the changes undergone by the steam 
during its entire action. This last area is ob- 
viously equal to the sum of the areas ABCD 
and EFGH. Figure 201, from Ewing (19) 
shows a pair of actual diagrams from a Woolf 
engine, the length of the diagrams representing 
the stroke of the pistons and not actual steam volumes. The low-pressure dia- 
gram has been reversed for convenience. Some expansion in the low-pressure 




Fig. 204. Art. 463, Prob. 31. —Dia- 
grams from "Woolf Engine. 



278 APPLIED THERMODYNAMICS 

cylinder occurs after the closing of the high-pressure exhaust valve at a. Some 
loss of pressure by wiredrawing iu the passages between the two cylinders is 
clearly indicated. 

464. Receiver Engine. In this more modern form the steam passes 
from the high-pressure cylinder to a closed chamber called the receiver, 
and thence to the low-pressure cylinder. The receiver is usually an inde- 
pendent vessel connected by pipes with the cylinders ; in some cases, the 
intervening steam pipe alone is of sufficient capacity to constitute a re- 
ceiver. Receiver engines may have the pistons coincident in phase, as in 
tandem engines, or opposite, as in opposed beam engines, or the cranks may 
be at an angle of 90°, as in the ordinary cross-compound. In all cases the 
receiver engine has the characteristic advantage over the Woolf type that 
the low-pressure cylinder need not receive steam during the whole of the 
working stroke, but may have a definite point of cut-off, and work in an 
expansive cycle. The distribution of work between the two cylinders, as 
will be shown, may be adjusted by varying the point of cut-off on the low- 
pressure cylinder (Art. 467). 

465. Drop. The fall of pressure occurring at the end of expansion 
is termed drop. Its thermodynamic disadvantage and practical necessity 
have been discussed (Arts. 418, 447). In a compound engine, drop in 
the high-pressure cylinder has the additional effect of seriously influenc- 
ing the amount of work done. With no such drop the combined ideal 
diagrams of a receiver engine would be precisely the same as that of a 
simple cylinder with the same amount of expansion. 

466. Combined Diagrams. Figure 205 shows the ideal diagrams from a tandem 
receiver engine. Along CD, as along CD in Fig. 203, expansion into the low- 
pressure cylinder is taking place. The cor- 
responding line on the low-pressure diagram 
is FG. At G the supply of steam is cut off: 
from the low-pressure cylinder, after which 
hyperbolic expansion occurs along GH. 
Meanwhile, the exhaust from the high-pres- 
sure cylinder is discharged to the receiver; 
and since a constant quantity of steam must 
now be contained in the decreasing space 

Fig. 205. Art. 466. -Combined Dia- between the piston and the cylinder and 
grams, Tandem Receiver Engine. receiver walls, some compression occurs, giv- 

ing the line DE. The pressure of the re- 
ceiver steam remains equal to that at E after the high-pressure exhaust valve 
closes (at E) and while the high-pressure cylinder continues the cycle along 
EABC. If the pressure at C exceeds that at E, then there will be some drop. 
As drawn, the diagram shows none. If cut-off in the low-pressure cylinder 




TANDEM RECEIVER ENGINE 279 

occurred later in the stroke, the line DE would be lowered, P c would exceed P E , 
and drop would be shown. 

An important advantage of the receiver engine is here evident. The intro- 
duction of cut-off in the low-pressure cylinder raises the lower limit of tempera- 
ture in the high-pressure cylinder from D in Fig. 203 to D in Fig. 205. This 
reduced range of temperature decreases cylinder condensation. 

467. Adjustment of Work. Figure 206 shows the diagrams as 
they appear with drop. Now if cut-off in the low-pressure cylinder 
be made to occur a little earlier in the stroke, the pressures at D 
and along the compression path DE 
would be increased, and the work area 
of the high-pressure cycle would be de- 
creased. The initial pressure in the low- 
pressure cylinder (which depends upon 
P E as well as P q) would be increased. 
The tendency toward a reduction of area 

of the low-pressure cycle by earlier cut- Fig. 206. Art. 467. — Receiver En- 

off is more than offset by the increased rop " 

initial pressure. The fact is that the total work of the engine is 
scarcely affected by a change in low-pressure cut-off. The low-pres- 
sure work area increases to almost precisely the same extent that 
the high-pressure area decreases. We have thus the peculiar re- 
sult that with earlier cut-off the low-pressure cylinder performs a 
greater proportion of the total work. Earlier cut-off decreases drop. 
The problem of compound engine design is to adjust the cylinder 
and receiver volumes and the point of low-pressure cut-off so that 
the desired amount of drop may be secured along with practically 
equal distribution of work between the two cylinders. 

468. Assumptions. In some cases, the cylinders are so proportioned as to 
make the range of temperatures the same in each. This usually involves the 
performance of very nearly equal amounts of work; the equalization of work 
areas is the more usual aim. The question of the desirable amount of drop will 
be considered later. For the present, we will assume it to be zero. In some 
marine engines, with valve gears which involve a rather late low-pressure cut-off 
at running speeds, the desired flexibility cannot be obtained without a consider- 
able amount of drop between the cylinders. 

469. Application to Tandem Compound. In Fig. 207, let A BCD be a portion 
of the indicator diagram of the high-pressure cylinder of a tandem receiver 



280 



APPLIED THERMODYNAMICS 



engine, release occurring at C. At this point, the whole volume of steam consists 
of that in the receiver plus that in the high-pressure cylinder. Let the receiver 
volume be represented by the distance CX. Then the hyperbolic curve XY may 

represent the expansion of the 
steam between the states C and 
D, and by deducting the constant 
volumes CX, LR, MZ, etc., we ob- 
tain the curve CG, representing 
the expansion of the steam in the 
two cylinders. For no drop, the 
pressure at the end of compression 
into the receiver must be equal 
to that at C. We thus find the 
point E, and draw EF, the ad- 
mission line of the low-pressure 
cylinder, such that ac + ad — ae, 
etc.; the abscissa of cC being to 
that of Ed in the same ratio as 
that of the respective cylinder 
volumes. By plotting ED we 
find the point D at its intersection with CD. A horizontal projection from D 
to EF gives F. The point F is then the required point of cut-off in the low- 
pressure cylinder. The diagram EFSHI may be completed, the curve FS being 
hyperbolic. 




Fig. 207. Art. 469. — Elimination of Drop, Tandem 
Receiver Engine. 



470. Cranks at Right Angles. In Fig. 208, let abC be a portion of the high- 
pressure diagram, release occurring at C. Communication is now opened with the 
receiver. Let the receiver volume be laid off as Cd, and let de be a hyperbolic 
curve. Then the curve Cf, the volume of which at any pressure is Cd less than 
that of de, represents the path in the high-pressure cylinder. This continues until 
admission to the low-pressure cylinder occurs at g. The whole volume of steam is 
now made up of that in the two cylinders and the receiver ; the volumes in the 
cylinders alone are measurable out to fC. In Fig. 209, lay off hi = IC and/& such 
that jk ~ hi is equal to the ratio of volumes of low- and high-pressure cylinder. 
At the point C of the cycle, the high -pressure crank is at i, the low-pressure crank 
90° ahead or behind it. When the high-pressure crank has moved from i to m, 
the volume of steam in that cylinder is represented by the distance hn, the low- 
pressure crank is at o and the volume of steam in the low-pressure cylinder is 
represented by pk. Lay off qr, in Fig. 208, distant from the zero volume line al 
by an amount equal to hn + pk. Draw the horizontal line is. Lay off tu = hi and 
to = us = pk. Then u is a point on the high-pressure exhaust line and v is a point 
on the low-pressure admission line. Similarly, we find corresponding crank posi- 
tions w and x, and steam volumes hy and zk, and lay off AB = hy + zk, Ac — hy, 
AD — cB = zk, determining the points c and D. The high-pressure exhaust line 
guc is continued to some distance below I. For no drop, this line must terminate 
at some point such that compression of steam in the high-pressure cylinder and 
receiver will make I the final state. At I the high-pressure cylinder steam volume 



CROSS-COMPOUND ENGINE 



281 



is zero ; all the steam is in the receiver. Let IE represent the receiver volume 
and EF a hyperbolic curve. Draw IG so that at any pressure its volumes are 
equal to those along EF, minus the constant volume IE. Then H, where IG 
intersects yuc, is the state of the high-pressure cycle at which cut-off occurs in 
the low-pressure cylinder. By drawing a horizontal line through H to intersect 
vD, we find the point of cut-off / on the low-pressure diagram. If we regard the 
initial state as that when admission occurs to the low-pressure cylinder, then at 

"KC* 

low-pressure cut-off the high-pressure cylinder will have completed the pro- 
portion of a full stroke. Modifications of this construction permit of determining 
the point of cut-off for no drop in triple or quadruple engines with any phase 
relation of the cranks. 



471. Analytical Method : Tandem Engine. Let the volume of the high-pressure 
cylinder be taken as unity, that of the receiver as R, that of the low-pressure cylinder 
as L. Let x be the fraction of its stroke completed by the low-pressure piston at 
cut-off, and let p be the pressure at release from the high-pressure cylinder, equal 
to the receiver pressure at the moment of admission to the low-pressure cylinder. 
The volume of steam at this moment is 1 + 7t : at low-pressure cut-off, it is 
1 + R + xL — x. If expansion follows the law joy = PV, andP be the pressure in 
the low-pressure cylinder at cut-off, 

P(\ + R)=P(1+R + xL-x),otP = p - l +R r . 

1 + R 4- xL — x 

The remaining quantity of steam in the high-pressure cylinder and receiver has 
the volume 1 — x + R, which, at the end of the stroke, will have been reduced to 
R. If the pressure at the end of the stroke is to be p, then 



pR=P(l-x + R) or P 

Combining the two values of P, we find 



1-x + R 



_ -R4-1 

X ~ RL + 1 

472. Cross-compound Engine. In this case, the fraction of the stroke completed 
at low-pressure cut-off is different for the two cylinders. Let X be the proportion 




Fig. 208. Arts. 470, 472, 473. — Elimination of Drop, Cross-compound Engine. 



282 



APPLIED THERMODYNAMICS 



of the high-pressure stroke occurring between admission and cut-off in the low- 
pressure cylinder. Proceeding as before, the volume of the steam at low-pressure 
admission is 0.5 + R, and that at low-pressure cut-off is 0.5 — X + R + xL. The 
volume of steam in the high-pressure cylinder and the receiver at the end of the 
high-pressure exhaust stroke is R ; the volume just after low-pressure cut-off occurs 
is 0.5 — X + R. The volume at the beginning of exhaust from the high-pressure 
cylinder is 1 + R. In Fig. 208, let the pressure at C and I be p ; let that at g be P. 



Then 



p(l + R) =P(0.5 + J R) or P = p 



0.5 + R 



Let the pressure at H be Q : then 

P(0.5 + 7T) = Q(0.5 
n - P(0. r o + R)q + R) 



X + R + xL), 
_ _ P (l + R) 



ButpR = Q(0.5 



(0.5-X + R + xL)(0.5 
-X+R), or p = Q 



R) 



(0.5 - X 



0.5- 
R) 



X + R + xL 

(l + fl)(0.5 -A' 




V 



R). 



Fig. 



209. Arts. 470, 472. — Crank Circles and Piston 
Displacements. 



J* 



R(0.5 - X + R + xL) 
whence, 

X=0.5 + R-xLR. (A) 

In Fig. 209, we have the crank 
circles corresponding to the 
discussed movements. If Ow 
and Ox are at right angles, 
then for a high-pressure pis- 
ton displacement Oy, we have 
the corresponding low-pres- 
sure displacement Jcz. If these 
displacements be taken as at 
low-pressure cut-off, then 

h i j k 

We may also draw OwP, PQ, 
and write X = — • In the 

triangles OPQ, Oxz, OQ = 
xz — jh . X, xz"-\- Oz"— Ox , and 



Equation (A), we find R (xL 



> whence X = Vx 
1) = 0.5 - Vx~^2 



x' 2 . Substituting this value in 
as the condition of no drop. 



473. Practical Modifications. The combined diagrams obtained from actual 
engines conform only approximately to those of Figs. 207 and 208. The receiver 
spaces are usually so large, in proportion to the volume of the high-pressure 
cylinder, that the fluctuations of pressure along the release lines are scarcely notice- 
able. The fall of pressure during admission to the low-pressure cylinder is, how- 
ever, nearly always evident. Marked irregularities arise from the angularity of the 



COMBINED DIAGRAMS 



283 



connecting rod. In assuming crank positions and piston displacements to corre- 
spond, we have tacitly assumed the rod to be of infinite length ; in practice, it seldom 
exceeds five or six times the length of the crank. The receiver volume is made 
from 1 to l 1 times that of the cylinder by which it is supplied. Its size has theo- 
retically no effect on the efficiency of the engine. We have assumed all expansion 
paths to be hyperbolic ; an assumption not strictly justified for the conditions con- 
sidered ; and we have ignored modifying influences due to clearance. Some designers, 
particularly in the case of marine engines, aim at equalizing the maximum pressures 
on cranks rather than at equalization of load; careful allowance must then be made 
for clearance and compression. 

474. Losses in Multiple-expansion Engines. Aside from those already discussed 
in connection with simple engines, the losses in a multiple-expansion engine 
include that due to pressure drop, if any, between the high-pressure cylinder 
and the receiver, and that due to friction in the intermediate passages. These 
are partially offset by superheating resulting from the wiredrawing. 



475. Combination of Actual Diagrams : Diagram Factor. Figure 210 shows the 
high- and low-pressure diagrams from a small compound engine. These are again 





b-> 




Fig. 210. 



Art. 475. — Compound Engine 
Diagrams. 



Fig. 211. 



Art. 475. — Compound Engine 
Diagrams Combined. 



shown in Fig. 211, in which the lengths of the diagrams are proportioned as are 
the cylinder volumes, the pressure scales are made equal, and the proper amounts 
of setting off for clearance (distances a and b) are regarded. The cylinder feed 
per single stroke was 0.0498 lb., the cushion steam in the high-pressure cylinder 
0.0074 lb., and that in the other cylinder 0.0022 lb. No single saturation curve 
is possible ; the line ss is drawn for 0.0572 lb. of steam, and SS for 0.0520 lb. As 
in Art. 453, we may obtain equivalent diagrams with the cushion steam eliminated. 
In Fig. 212, the single curve SS then represents saturation for 0.0498 lb. of steam. 
The areas of the diagrams are unaltered, and correctly measure the work done ; 
they may be transferred to the entropy plane as in Art. 454. The moisture present 
at any point during expansion is still represented by the distance cd, correspond- 
ing to the distance similarly marked in Fig. 211. In Fig. 213 this construction 
has been applied to a triple-expansion engine, the first diagram showing the 
action when un jacketed, and the second, when jackets are used. The drying- 
influence of the jackets is clearly shown. The ratio of the area of the combined 



284 



APPLIED THERMODYNAMICS 



actual diagrams to that of the Rankine cycle through the same extreme limits 
of pressure and with the same ratio of expansion is the diagram factor, the value 





Fig. 212. Art. 47n. — Combined Diagrams 
for Cylinder Feed. 



Fig. 213. 



UNJACKETED 



Art. 475. — Triple Engine 
Diagrams. 



of which may range from 0.70 upward, being higher than in simple engines having 
the same total ratio of expansion, but not higher than in the simple engines of 
ordinary practice (Art. 459). 

476. Compound Engine Capacity. If e be the real ratio of expansion in the 
high-pressure cylinder, and L the ratio of cylinder volumes, the total real ratio of 
expansion is E = eL. If i is the clearance in the high-pressure cylinder expressed 
as a fraction of the volume of that cylinder, and k is the apparent ratio of expan- 
sion therein, we may show that k = —^ — , E = The total real ratio of 

1 — ie 1 — i + ik 

expansion is thus independent of the point of cut-off on the low-pressure cylinder. It 
ranges usually from 10 to 25, increasing as the number of expansive cylinders 
increases. In compound engines it is most commonly 16. 

Given the same initial pressure and back pressure, total real ratio of expansion, 
and diagram factor, the low-pressure cylinder volume of a multiple-expansion engine 
is obviously the same as that of the simple engine cylinder of equal capacity. It is 
common practice to establish mean receiver pressures which will at normal load, 
without drop, give equal distribution of work between the cylinders. If the vari- 
ous computed mean effective pressures are then divided by the ratio of low-pressure 
cylinder volume to that of the cylinder under consideration, and the quotients 
added, we have the "mean effective pressure referred to the low-pressure cylinder." 
The capacity may be calculated from this and from the dimensions and piston 
speed of that cylinder. 

The size of the low-pressure cylinder being determined as S, that of ths high- 
pressure cylinder is —S, the minimum value of which is The value of E 

E E 

may be adjusted at will by varying the point of high-pressure cut-off, regardless of 
the cylinder ratio. From this standpoint, then, the size of the high-pressure 
cylinder is without influence on the efficiency. Non-condensing compound engines 
usually have a low-pressure cylinder from 3 to 4 times larger than the high-pres- 
sure cylinder. With condensing engines, the ratio is usually 4 to 6, increasing 
with the boiler pressure. In triple engines, the ratios are from 1 :2.0 : 2.0 up to 



DESIGN OF COMPOUND ENGINE 



285 



1 : 2.5 : 2.5 in stationary practice. In quadruple engines the ratios are successively 
from 2.0 to 2.5 : 1. The use of two-stage or compound expansion is common prac- 
tice everywhere. Triple and quadruple engines, in which much higher initial 
pressures are desirable, are used mostly in marine service. In. stationary applica- 
tions, a few of these high-stage engines are in use, with excellent results as to fuel 
economy ; but it is only where the cost of fuel or the load factor is high or capital 
charges low that they have to any considerable extent been found more profitable, 
commercially, than the compound engine. 

477. Specimen Design. Let the engine develop 1000 Ihp. at 100 r. p. m,, 
with pressure limits of 150 and 2 lb. absolute and a ratio of expansion of 16, 
the piston speed being 800 ft. per minute. 

In a simple engine, the m. e. p. would be (Art. 446) ^ "^ ^ e — ^ — 2 

= 33.5, and the stroke - 8 



16 



4 feet or A8 inches. We will ignore the 
2x100 8 

diagram factor in order to more rigorously compare sizes ; the area of the 
cylinder of the simple engine is then (33,000 x 1000) -=- (33.5 x 800)= 1230 
square inches. 

In the compound engine, let the cylinder ratio be first established, say 
as 4. The mean effective pressure of the combined diagrams is 33.5. If 
we assign half of this to the low-pressure cylinder, its area must be (500 X 
"33.5 



33,000)- 



x800 



= 1230 square inches, precisely that of the simple 



cylinder. The m.e.p. in the high-pressure cylinder referred to the loiv-pres- 

335 . 33 5 

sure cylinder (Art. 476) is also K —^—: its actual m. e.p. is then ^— x4 = 67, 

and its area is (33,000 x 500) -v- (67 x 800) = 307\ square inches : or, more 

1230 
briefly, = 307^. This gives an engine in which the work distribu- 

tion with no drop may be unequal.* If actual diagram factors are intro- 
duced, the low-pressure cylinder of the compound will differ somewhat in 
size from the cylinder of the equivalent p 
simple engine. 

478. Governing Compound Engines. It 
has been shown (Ait. 467) that earlier cut- 
off on the low-pressure cylinder relieves the 
high-pressure cylinder of some of its propor- 
tion of the load. Figure 214 shows further 
that delayed cut-off on die high-pressure cyl- 
inder greatly increases the work done in the 
low-pressure cylinder, while only slightly 



i — r -1 
\ t 

\ v ° 
-Vd 


a 

z 
j 
o 

!^ 

Q. ^^^»^_^ "" — _ 
I ' ' 




f 


""""~"n 









Fig. 214. 



Art. 478.— Effect of Low- 
pressure Cut-off. 



* High cylinder ratios, with equal work distribution, are possible only when the 
total number of expansions is high. It is, of course, permissible to design the engine 
so that each cylinder does half the work. See problem 27, page 314. 



286 APPLIED THERMODYNAMICS 

increasing its own work area. When the load increases in an engine which is gov- 
erned by adjustment of the high-pressure cut-off only, equality of work distribu- 
tion becomes impossible. For economy, the governor should control the cut-off 
on both cylinders, making it later on both as the load increases, but not in the 
same proportion. 

Variation of cut-off in the low-pressure cylinder permits of adjustment of the 
division of work between the cylinders, irrespective of the sizes ; but absence of 
drop is simultaneously possible only when the cylinder ratios are correct. Adjust- 
ment of low pressure cut-off to eliminate drop, in badly proportioned cylinders, 
results in an unequal distribution of work. 

To summarize : the power of the engine is varied by varying the 
high-pressure cut-off; wide ranges of capacity are obtainable only 
when the high-pressure cylinder is comparatively large : the distri- 
bution of the work is kept uniform by varying the low-pressure cut- 
off ; and this results in a loss of efficiency due to the excessive drop 
unless the cylinder proportions are right. 

479. The Drop Controversy. Thus far, we have treated the subject from the 
standpoint that maximum efficiency is attained with a zero drop in pressure at 
high-pressure release. This is the orthodox view, maintained in this country by 
many engineers, and almost universally followed in European practice. Some 
authorities have contended that a limited amount of drop is both practically and 
thermodynamically desirable (21). From Art. 447, it is obvious that in a single 
cylinder, expansive ratios exceeding certain limits become undesirable on account 
of excessive cylinder condensation: in such cylinders, a constant volume drop at 
the end of the stroke is always permitted. In a compound engine, drop decreases 
the diagram factor of the combined diagram : and it has been usually regarded as 
objectionable on any but the last cylinder of the series. The aim of designers has 
been to make the actual expansion line coincide with the hyperbolic curve as 
closely as possible ; and for this reason the harmful influence of drop has possibly 
been overemphasized, and the argument in its favor disregarded. There is, in 
fact, a special argument for drop in multiple-expansion engines, from the fact 
that unresisted expansion leads to a drying of the steam, which exerts a beneficial 
effect in the succeeding cylinder. 

480. Intermediate Compounds. Tests by Rockwood (22) of a triple 
engine in which the intermediate cylinder was cut out, permitting of run- 
ning the high- and low-pressure cylinders as a compound with the high 
cylinder ratio of 5.7 to 1, give the surprising result that with the same 
initial pressure and expansive ratio, the compound was more economical 
than the triple. This was a small engine, with large drop. The pointing 
out of the fact that the conditions were unduly favorable to the compound 
as compared with the triple did not explain the excellent economy of the 
former as compared with all engines of its class. Somewhat la^er, excep- 



REHEATERS 



287 



tionally good results were obtained by Barrus (23) with a compound 
engine having the extraordinary cylinder ratio of 7.2 : 1.0. Thurston, 
meanwhile, experimented in the same manner as Rock wood, determining, 
in addition, the economy of the high-pressure and intermediate cylinders 
when run together as a compound. There were thus two compounds of 
ratios 3.1 : 1 and 7.13 : 1 and a triple of ratio 1 : 3.1 : 2.3, available for test. 
The results showed the 7.1 compound to be much better than the 3.1, but 
less economical than the triple (24). As the ratio of expansion decreased, 
the economy of the intermediate compound closely approached that of the 
triple ; and at a very low ratio it would probably have equaled it. The 
deduction is that the triple engine shows the efficiency to be expected 
when the ratio of expansion is high, as it should be for a triple engine, 
but that a high ratio (" intermediate ") compound may far surpass an ordi- 
nary compound in economy. Ordinary compound engines usually have 
the high- press ure cylinders too large, a result of the aim toward excessive 
overload capacity. 

481. Reheating. A considerable gain in economy is attained by 
superheating the steam during its passage through the receiver, by 
means of pipe coils supplied with high-pressure steam from the boiler. 
The argument in favor of reheating is the same as that for the use of 
superheat in any cylinder (Art. 442). It is not surprising, therefore, that 




WITHOUT REHEATERS 




TH REHEATERS 



Figs. 215 and 21G. Art. 481. —Effect of Reheatim 



the use of reheaters is only profitable when a considerable amount of 
superheating — not less than 100° F. — is effected. Reheating was formerly 
unpopular, probably because of the difficulty of securing a sufficient 
amount of superheat when saturated steam was used in the receiver 
coils. With superheated steam, this difficulty is obviated. Reheating 
greatly increases the capacity as well as the economy of the cylinders, 
as is shown by Figs. 215 and 216, representing the PV and TN diagrams 
of a 760-hp. cross-compound engine (25). 

482. Superheat and Jackets. Since multiple expansion itself decreases 
cylinder condensation, these refinements cannot be expected to lead to such 



288 



APPLIED THERMODYNAMICS 



large economies as in simple engines. Moderately superheated steam has, 
however, given excellent results, eliminating cylinder condensation so per- 
fectly as to permit of wide ranges of expansion without loss of economy 
and thus making the efficiency of the engine, within reasonable limits, 
almost independent of its load. With efficient superheating and reheat- 
ing, jackets are of little value. 

483. Binary Vapor Engine. This was originated by Du Tremblay in 1850 
(26). The exhaust steam from a cylinder passed through a vessel containing 
coils filled with ether. The steam being at a temperature of almost 250° F., 
while the atmospheric boiling point of ether is 94° F., the latter was rapidly 
vaporized at a considerable pressure, and was then used for performing work in 
a second cylinder. Assuming the initial temperature of the steam to have been 
320° F., and the final temperature of the ether 100° F., the ideal efficiency should 
thus be increased from 

320-250 = a09to 320-100 
320 + 460 320 + 460 

a gain of over 200 per cent. The advantage of the binary vapor principle arises 
from the low boiling point of the binary fluid. This permits of a lower tempera- 
ture of heat emission than is possible with water. Binary engines must be run 
condensing. Since condensing water is generally not available at temperatures 
below 60° or 70° F., the fluid should be one which may be condensed at these tem- 
peratures. Ether satisfies this requirement, and gives, at its initial temperature 
of, say, 250° F., a working pressure not far from 150 lb. On account of its high 
boiling point, however, its pressure is less than that of the atmosphere at 70° F., 
and an air pump is necessary to discharge the condensed vapor from the condenser 
just as is the case with condensing steam engines. Sulphur dioxide has a much 
lower boiling point, and may be used without an air 
pump : but its pressure at 250° would be excessive, and 
the best results are secured by allowing the steam cylinder 
to run condensing at a final temperature as low as pos- 
sible ; at 104° F., the pressure of sulphur dioxide is only 
90.3 lb. The best steam engines have about this lower 
temperature limit; the maximum gain due to the use of a 
binary fluid cannot exceed that corresponding to a reduc- 
tion of this temperature to about 60° or 70° F., the usual 
temperature of the available supply of cooling water. 

The steam-ether engines of the vessel Bresil operated 
at 43.2 lb. boiler pressure and 7.6 lb. back pressure of 
ether. The cylinders were of equal size, and the mean 
effective pressures were 11.6 and 7. 1 lb. respectively. The 
coal consumption was brought down to 2.44 lb. per 
Ihp.-hr. ; a less favorable result than that obtainable from 
good steam engines of that time. Several attempts have 
been made to revive the binary vapor engine on a small scale ; the most important 
recent experiments are those of Josse (27), on a 200-hp. engine using steam 




z m 

•Fia.217. Art. 483, Prob. 
59. — Binary Vapor En- 
gine. 



THE INDICATOR 289 

at 160 lb. pressure and 200° of superheat, including three cylinders. The first 
cylinders constitute an ordinary compound-condensing steam engine, a vacuum 
of 20 to 25 in. of mercury being maintained in the low-pressure cylinder by the 
circulation of sulphur dioxide in the coils of a surface condenser. The dioxide 
then enters the third cylinder at from 120 to 180 lb. pressure and leaves it at 
about 35 lb. pressure. The best result obtained gave a consumption of 167 
B. t. u. per Ihp. per minute, a result scarcely if ever equaled by a high-grade 
steam engine (Art. 550). The ideal entropy cycle for this engine is shown in 
Fig. 217, the two steam cylinders being treated as one. The steam diagram is 
abcde, and the heat delivered to the sulphur dioxide vaporizer is aerm. This 
heated the binary liquid along hi and vaporized it along if, giving the work area 
hifg. The different liquid lines and saturation curves of the two vapors should be 
noted. The binary vapor principle has been suggested as applicable to gas en- 
gines, in which the temperature of the exhaust may exceed 1000° F. 

Engine Tests 

484. The Indicator. Two special instruments are of prime importance in 
measuring the performance of an engine. The first of these is the indicator, one 

of the secret inventions of Watt (28), which . -v 

shows the action of the steam in the cylinder. QZ^^ "■^^7 

Some conception of the influence .of this device |V J| 

on progress in economical engine operation may ^=>— — LI 

be formed from the typically bad and good dia- 
grams of Fig. 218. The indicator furnishes a 
method for computing the mean effective pres- 
sure and the horse power of any cylinder. 

Figure 219 shows one of the many common 
forms. Steam is admitted from the engine cylin- 
der through 6 to the lower side of the movable FlG " 218> Arts " 484 ' m -~ Good 

• j. o rr-, a *.- £ ^i and Bad Indicator Diagrams, 

piston 8. The fluctuations of pressure in the 

cylinder cause this piston to rise or fall to an extent determined by the stiffness 
of the accurately calibrated spring above it. The piston movements are trans- 
mitted through the rod 10 and the parallel motion linkage shown to the pencil 
23, where a perfectly vertical movement is produced, in definite proportion to 
the movement of the piston 8. By means of a cord passing over the sheaves 
37, 27, a to-and-fro movement is communicated from the crosshead of the engine 
to the drum, 24. The movements of the drum, under control of the spring, 31, 
are made just proportional to those of the piston; so that the coordinates of the 
diagram traced by the pencil on the paper are pressures and piston movements. 

4. 

485. Special Types. Various modifications are made for special applications. 
For gas engines, smaller pistons are used on account of the high pressures ; springs 
of various stiffnesses and pistons of various areas are employed to permit of accu- 
rately studying the action at different parts of the cycle, safety stops being pro- 
vided in connection with the lighter springs. The Mathot .instrument, for 
example, gives a continuous record of the ignition lines only of a series of sue- 



290 



APPLIED THERMODYNAMICS 



cessive gas engine diagrams. " Outside-spring " indicators are a recent type, in 
which the spring is kept away from the hot steam. The Ripper mean-pressure 
indicator (29-) is a device which shows continuously the mean effective pressure 
in the cylinder. Instruments are often provided with pneumatic or electrical 
operating mechanisms, permitting one observer to take exactly simultaneous dia- 
grams from two or more cylinders. Indicators for ammonia compressors must 
have all internal parts of steel; special forms are also constructed for heavy hy- 




Fig. 219. Art. 484. — Crosby Steam Engine Indicator. 



draulic and ordnance pressure measurements. For very high speeds, in which the 
inertia of the moving parts would distort the diagram, optical indicators are used. 
These comprise a small mirror which is moved about one axis by the pressure and 
about another by the piston movement. The path of the beam of light is pre- 
served by photographing it. Indicator practice constitutes -an art in itself; for 
the detailed study of the subject, with the influence of drum reducing motions, 
methods of calibration, adjustment, piping, etc., reference should be made to such 
works as those of Carpenter (30) or Low (31). In general, the height of the dia- 
gram is made of a convenient dimension by varying the spring to suit the maxi- 
mum pressure ; and accuracy depends upon a just proportion between (a) the 
movements of the drum and the engine piston and (b) the movement of the indi- 
cator piston and the fluctuations in steam pressure. 



INDICATOR DIAGRAMS 



291 




\ 




486. Measurement of Mean Effective Pressure. This may be accomplished 
by averaging a large number of equidistant ordinates across the diagram, or, 
mechanically, by the use of the planimeter (32). In usual practice, the indicator is 
either piped, with intervening valves, to both ends of the cylinder, in which case a 
pair of diagrams is obtained, as in Fig. 218, one cycle after the other, representing 
the action on each side of the piston ; or two diagrams are obtained by separate 
indicators. In order that the diagrams may be complete, the lines ab, representing 
the boiler pressure, cd, of atmospheric pressure, and efoi vacuum in the condenser, 
should be drawn, together with the line of zero volume ea, determined by measur- 
ing the clearance, and the hyperbolic curve ij, constructed as in Art. 92. The 
saturation curve gh for the amount of steam actually in the cylinder is sometimes 
added. 

487. Deductions. By taking a " full-load" card, and then one with the ex- 
ternal load wholly removed, the engine overcoming its own frictional resistance 
only, we at once find the me- 
chanical efficiency, the ratio of 
power exerted at the shaft to 
power developed in the cylin- 
der; it is the quotient of the 
difference of the two diagrams 
by the former. By measur- 
ing the pressure and the vol- 
ume of the steam at release, 
and deducting the steam pres- 
ent during compression, we 
may in a rough way com- 
pute the steam consumption 
per Ihp.-hr., on the assumption 
that the steam is at this point 
dry ; and, as in Art. 500, by 
properly estimating the per- 
centage of wetness, we may 
closely approximate the actual 
steam consumption. 

Some of the applications 
of the indicator are suggested 
by the diagrams of Fig. 220. 
In a, we have admission oc- 
curing too early; in b, too 
late. Excessively early cut-off 
is shown in c ; late cut-off, with 
excessive terminal drop, in d. 
Figure e indicates too early f ig 220 
release ; the dotted curve 
would give a larger work area ; 

in f, release is late. The bad effect of early compression is indicated in g ; late com- 
pression gives a card like that of h, usually causing noisiness. Figure i shows exces- 









D^ 






Art. 487. — Indicator Diagrams and Valve 
Adjustment. 



292 APPLIED THERMODYNAMICS 

sive throttling during admission ; j indicates excessive resistance during exhaust 
which may be due to throttling or to a poor vacuum. The effect of a small supply 
pipe is shown in k; in which the upper line represents a diagram taken with the 
indicator connected to the steam chest. The abrupt rise of pressure along BC is 
due to the cutting off of the flow of steam from the steam chest to the cylinder. 
Figure I shows the form of card taken when the drum is made to derive its mo- 
tion from the eccentric instead of the crosshead. This is often done in order to 
study more accurately the conditions near the end of the stroke when the piston 
moves very slowly, while the eccentric moves more rapidly. Figure m is the cor- 
responding ordinary diagram, and the two diagrams are correspondingly lettered. 
Figure n is an excellent card from an air compressor; o shows a card from an air 
pump with excessive port friction, particularly on the suction side. Figure p 
shows what is called a stroke card, the dotted line representing net pressures on 
the piston, obtained by subtracting the back pressure as at ab from the initial 
pressure ac, i.e. by making dc = ab. Figure q shows the effect of varying the 
point of cut-off; r, that of throttling the supply. Negative loops like that of g 
must be deducted from the remainder of the diagram in estimating the work done. 

488. Measurement of Steam Quality. The second special instrument used in 
engine testing is the steam calorimeter, so called because it determines the percent- 
age of dryness of steam by a series of heat measurements. Carpenter (33) classi- 
fies steam calorimeters as follows : 



Calorimeters 
Jet 



(a) Condensing 



Barrel or tank 
Continuous 
f Barrus — Continuous 



to 



(b) Superheating 



Surface < Hoadley 
I Kent 
f External — Barrus 
{ Internal — Peabody 



(c) Direct {Separator 

(C) " UneCt {Chemical 

489. Barrel or Tank Calorimeter. The steam is discharged directly into an 
insulated tank containing cold water. Let W, to be the weights of steam and 
water respectively, t, h the initial and final temperatures of the water, correspond- 
ing to the heat quantities h, hi) and let the steam pressure be P , corresponding 
to the latent heat L and heat of liquid ho, the percentage of dryness being x . 
The heat lost by the steam is equal to the heat gained by the water ; or, the steam and 
w T ater attaining the same final temperature, 

W (xoLo + ho — hi) = iv(hi — h), whence x = -^ ! d=- 

WL 

The value of W is determined by weighing the water before and after the mix- 
ture. The radiation corrections are large, and any slight error in the value of W 



CALORIMETERS 



293 



greatly changes the result; this form of calorimeter is therefore seldom used, its 
average error even under the best conditions ranging from 2 to 4 per cent. Some 
improvement is possible by causing condensation to become continuous and tak- 
ing the weights and temperatures at frequent intervals, as in the "Injector" or 
" Jet Continuous " calorimeter. 

490. Surface-condensing Calorimeter. The steam is in this case condensed 
in a coil ; it does not mingle with the water. Let the final temperature of the 
steam be h, its heat contents 7j 2 ; then 

wli x + Wh 2 - Kh - Wh 



W(xqLq + ho — hi) = w (hi — h) and xo 



WL 



More accurate measurement of W is possible with this arrangement. In the 
Hoadley form (34) a propeller wheel was used to agitate the water about the coils; 
in the Kent instrument, arrangement was made for removing the coil to permit 
of more accurately determining W. In that of Barrus, the flow was continuous 
and a series of observations could be made at short intervals. 

491. Superheating Calorimeters. The Peabody throttling calorimeter 
is shown in Fig. 221 ; steam entering at b through a partially closed valve 
expands to a lower steady pressure in A and then flows into the atmos- 
phere. Let L 0f h , x be the condition at b, and assume the steam to be 
superheated at A, its temperature being T, t being the 
temperature corresponding to the pressure p, and the cor- 
responding total heat at saturation H. Then, the total heat 
at b equals the total heat at A, or 

(x L + ho)=H+Jc(T-f), 

w r here k is the mean specific heat of superheated steam 
at the pressure p between T and t ; whence 

H+k(T-t)-h 

JL — 

If we assume the pressure in A to be that of the atmos- 
phere, £"=1150.4, and superheating is possible only when 
x L -f- h exceeds 1150.4. For each initial pressure, then, 
there is a corresponding minimum value of x beyond 
which measurements are impossible; thus, for 200 lb., 
L = 843.2, h = 354.9, and x Q (minimum) is 0.94^ Aside 
from this limitation, the throttling calorimeter is exceed- 
ingly accurate if the proper calibrations, corrections, and methods of 
sampling are adopted. In the Barrus throttling calorimeter, the valve at 
b is replaced by a diaphragm through which a fine hole is drilled, and the 
range of x values is increased by mechanically separating some of the 
moisture. The same advantage is realized in the Barrus superheating 
calorimeter by initially and externally heating the sample of steam. The 




Fig. 221. Art. 491. 
— Superheat- 
ing Calorimeter. 



294 



APPLIED THERMODYNAMICS 



amount of heat thus used is applied in such a way that it may be ac- 
curately measured. Let it be called, say, Q per pound. Then 

x L + h +Q = H+k(T-t),ari(lx = H + k ( T -^- h »-Q . 



492. Separating Calorimeters. The water and steam are mechanically sepa- 
rated and separately weighed. In Fig. 222, steam enters, through 6, the jacketed 
chamber shown. The water is intercepted by the cup 
14, the steam reversing its direction of flow at this 
point and entering the jacket space 7, 4, whence it is 
discharged through the small orifice 8. The water ac- 
cumulates in 3, its quantity being indicated by the 
gauge glass 10. The quantity of steam flowing is de- 
termined by calibration for each reading of the gauge 
at 9. The instrument is said to be fairly accurate un- 
less the percentage of moisture is very small. The 
steam may be, of course, run off, condensed, and 
actually weighed. 

493- Chemical Calorimeter. This depends for its 
action on the fact that water will dissolve certain salts 
(e.g. sodium chloride) which are insoluble in dry 
steam. 

494- Electric Calorimeter. The Thomas superheat- 
ing and throttling instrument (35) consists of a small 
soapstone cylinder in which are embedded coils of 
German silver wire, constituting an electric heater. 
This is inserted in a brass case through which flows 
a current of steam. The electrical energy correspond- 
ing to heat-augmentation to any superheated condition being known, say, as 
E B. t. u. per pound (1 B. t. u. = 17.59 watts per minute), we have, as in Art. 491, 




Fig. 222. Art. 492.— 
rating Calorimeter 



x L Q + h 4- E = H 4- k(T - t), whence x = 



H+l'(T-i) 



495. Engine Trials: Heat Measurement. AYe may ascertain the heat 
supplied in the steam engine cycle either by direct measurement, or by 
adding the heat equivalent of the external work done to the measured amount 
of heat rejected. In the former case the amount of water fed to the boiler 
must be determined, by weighing, measuring, or (in approximate work) by 
the use of a water meter. The heat absorbed per pound of steam is ascer- 
tained from its temperature, quality, and pressure, and the temperature of 
the water fed to the boiler. In the latter case, the steam leaving the 
engine is condensed and, in small engines, weighed ; or in larger engines, 
determined by metering or by passing it over a weir. This latter of the 
two methods of testing has the advantage with small engines of greater 



ENGINE TEST 



295 



accuracy and of giving accurate results in a test of shorter duration. Where 
the engine is designed to operate non-condensing, the steam may be con- 
densed for the purposes of the test by it passing it over coils exposed to 
the atmosphere, so that no vacuum is produced by the condensation. If 
jackets are used, the condensed steam from them must be trapped off and 
weighed. This water would ordinarily boil away when discharged at 
atmospheric pressure, so that provision must be made for first cooling it. 

496. Heat Balance. By measuring both the heat supplied and that rejected, as 
well as the work done, it is possible to draw up a debit and credit account show- 
ing the use made of the heat and the unaccounted for losses. These last are due 
to the discharge of water vapor by the air pump, to radiation, and to leakage. 
The weight of steam condensed may easily be four or five per cent less than that 
of the water fed to the boiler. Let ff, h, be the heat contents of the steam and 
the heat in the boiler feed water respectively; the heat absorbed per pound is 
then H — h. Let Q be the heat contents of the exhausted steam (measured 
above the feed water temperature) and W the heat equivalent of the work done 
per pound. Then for a perfect heat balance, H — h = Q + W. In practice, W 
is directly computed from the indicator diagrams ; H and Q must be corrected 
for the quality of steam as determined by the calorimeter or otherwise. 

497. Checks; Codes. Where engines are used to drive electrical generators 
the measurement of the electrical energy gives a close check on the computation 
of indicated horse power. In locomotive trials a similar check is obtained by 
comparison of the drawbar pull and speed (36). In turbines, the indicator 
cannot be employed; measurement of the mechanical power exerted at the shaft 
is effected by the use of the friction brake. Standard codes for the testing of 
pumping* engines (37), and of steam engines generally (38), have been developed 
by the American Society of Mechanical Engineers. 

498. Example of an Engine Test. Figure 224, from Hall (39), gives 
the indicator diagrams from a 30 and 56 by 72-in. compound engine at 



The piston rods were 4 



diameter. The boilei 




Fig. 224. Arts. 498, 499, 500. — Indicator Cards from Compound Engine. 



296 APPLIED THERMODYNAMICS 

pressure was 124.0 lb. gauge: the pressure in the steam pipe near the 
engine, 122.0 lb. The temperature of jacket discharge was 338° F. The 
conditions during the calorimetric test of the inlet steam were P = 122.08 
lb. gauge, T=z 302.1° F. (Art. 491), pressure in calorimeter body (Fig. 221), 
11.36 lb. (gauge). The net weight of boiler feed water in 12 hours was 
231,861.7 lb. ; the weight of water drained from the jackets, 15,369.7 lb. 

From the cards, the mean effective pressures were 44.26 and 13.295 
lb. respectively ; and as the average net piston areas were 697.53 and 
2452.19 square inches respectively, the total piston pressures were 44.26 
X 697.53=30872.7 and 13.295 x 2452.19=32601.9 lb. respectively. These 
were applied through a distance of 

\\ X 2 x 58 = 696 feet per minute ; 

whence the indicated horse power was 

(30872.7 + 32601.9) x 696 = 1338 62 
33000 ' ' 

From Art. 491, x L +h = H+Jc (T- t), or in this case, 866.5 x + 322.47 
= 1155.84 + 0.48* (302.1-242.3) whence x = 0.995. The weight of 
cylinder feed was 231,861.7-15,369.7 = 216,492.0 lb. At its pressure of 
136.7 lb. absolute, Z = 866.5, ft = 322.4. For the ascertained dryness, the 
total heat per pound, above 32°, is 322.4 + (0.995x866.5) = 1184.5 B. t. u. 
The heat left in the steam at discharge from the condenser (at 114° F.) 
was 82 B. t. u. ; the net heat absorbed per pound of cylinder feed was 
then 1184.5 — 82.0 = 1102.5 ; for the total weight of cylinder feed it was 
1102.5 x 216,492 = 238,682,430 B. t. u. The total heat in one pound of 
jacket steam was also 1184.5 B. t. u. This was discharged at 338° F. 
(Ji = 308.8), whence the heat utilized in the jackets was 1184.5 — 308.8 
= 875.7 B. t. u. (The heat discharged from both jackets and cylinders 
was transferred to the boiler feed water, the former at 338°, the latter at 
114° F.) The supply of heat to the jackets was then 875.7 X 15,369.7 
=13,459,246.29 B. t. u : the total to cylinders and jackets was this quan- 
tity plus 238,682,430 B. t. u., or 252,141,676.29 B. t. u. Dividing this by 
60 x 12 we have 350,196.77 B. t. u. supplied per minute. 

499. Statement of Results. We have the following : 

(a) Pounds of steam per Ihp.-hr. = 231,861.7 -=- 12 -- 1338.62 = 14.43. 
(This is the most common measure of efficiency, but is wholly 
unsatisfactory when superheated steam is used.) 

* Value taken for the specific heat of superheated steam. 



STEAM CONSUMPTION FROM DIAGRAM 297 

(b) Pounds of dry steam per Ihp.-hr. = 14.43 x 0.995 * = 14.36. 

(c) Heat consumed per Ihp. per minute == 350,196.77 h- 1338.62 = 261.61 

B. t. u. 

(d) Thermal efficiency = ^f^-s- 261.61 = 0.1621. 

(e) Work per pound of steam = 25 ^l^ 29 X 0.1621 = 176 B. t. u. 

(/)Carnot efficiency =J^=^ = 0.293. 



(g) Clausius efficiency (Art. 409), with dry steam, 

/oKi 90 1W1. 866^ ^o^i 810.82 
(351.22-114)^1 4-^^j -573.6 log e ^j 

351.22-114 + 866.5 

(ft) Ratio of (d) -*- (gr) = 0.1621 -~ 0.265 = 0.61. 



= 0.265. 



500. Steam Consumption from Diagram. The inaccuracy of such estimates 
will be shown. In the high-pressure cards, Fig. 224, the clearance space at each 
end of the Cylinder was 0.932 cu. ft. The piston displacement per stroke on the side 
opposite the rod was 706.86 x 72 -=- 1728 = 29.453' cu. ft. ; the cylinder volume 
on this side was 29.453 + 0.932 = 30.385 cu. ft. The length of the correspond- 
ing card (a) is 3.79 in. ; the clearance line be is then drawn distant from the 
admission line 

3.79 x -M?i = 0.117 in. 
29.453 

At d, on the release line, the volume of steam is 30.385 cu. ft., and its pressure is 
31.2 lb. absolute. From the steam table, the weight of a cubic foot of steam at 
this pressure is 0.076362 lb. ; whence the weight of steam present, assumed dry, is 
0.076362 x 30.385 = 2.3203 lb. At a point just after the beginning of compres- 
sion, point e, the volume of steam expressed as a fraction of the stroke plus the 
clearance equivalent is 0.517 -=- 3.907 = 0.1321, 3.907 being the length bg in inches. 
The actual volume of steam at e is then 0.1321 x 30.385 = 4.038 cu. ft., and its 
pressure is 28.3 lb. absolute, at which the specific weight is 0.069683 lb. The 
weight present at e is then 4.038 x 0.069683 = 0.280 lb. The net weight of steam 
used per stroke is 2.3203 - 0.280 = 2.04031b., or, per hour, 2.0403 x 58 x 60 = 7090 
lb., for this end of the cylinder only. For the other end, the weight, similarly 
obtained, is 7050 lb.; the total weight is then 14,140 lb. The horse power 
developed in the high-pressure cylinder is 650, and the cylinder feed per Ihp.-hr. 
from high-pressure diagrams is 21.8 lb. The same process may be applied to the 
low-pressure diagrams. It is best to take the points d and e just before the begin- 
ning of release and after the beginning of compression respectively. The method 

*The factor 0.995 does not precisely measure the ratio of energy in the actual 
steam to that in the corresponding weight of dry steam, but the correction is usually 
made in this way. 



298 APPLIED THERMODYNAMICS 

is widely approximate, but may give results of some value in the absence of a 
standard trial, if the quality of steam at release and compression is known (Arts. 
448, 440). 

501. General Analysis. Let A, a represent the areas of the two sides of the 
piston, P, p the corresponding mean effective pressures, S the length of the stroke, 
and R the number of revolutions per minute. The indicated horse power is, then, 

(AP + ap)SR 
33000 

Let B, b denote the ratios of volume at release to total cylinder volume, W, w, 
the corresponding specific weights, T, t the ratios of volume at compression to 
total cylinder volume, and V, v the corresponding specific weights at that point ; 
then, if C, c denote the clearance volumes, the volumes of steam at release are 
B(AS + C) and b(aS + c) ; the weights are WB(AS + C) and wb(aS + c) ; the 
volumes at compression are T(AS + C) and t(aS + c) ; the weights there are 
VT(AS + C) and vt(aS + c) ; the weight of cylinder feed per revolution is then 
WB(AS + C) + wb(aS + c) - VT(AS + C) - vt(aS + c) ; or, per hour, it is 60 R 
times this. The quotient of this expression by that given for horse power gives 
the steam consumption per indicated horse power hour, directly derived from the 
cards; and if C, c be expressed as functions of the area and stroke, say as aAS, 
in which a is the ratio of clearance to piston displacement, the values of A, S, and R 
cancel out so that no information is necessary other than that given by the diagrams 
themselves. 

502. Measurement of Rejected Heat. A common example is in tests in 
which the steam is condensed by a jet condenser (Art. 584). In a test 
cited by Ewing (40), the heat absorbed per revolution measured above the 
temperature of the boiler feed was 1551 B. t. u. ; that converted into work 
was 225 B. t. u. The exhaust steam was mingled with the condensing 
water, a combined weight of 51.108 lb. being found per revolution. The 
temperature of the entering water was 50° F., that of the discharged mix- 
ture was 73.4° F., and the cylinder feed amounted to 1.208 lb. per revolu- 
tion. The temperature of the boiler feed water was 59° F. We may 
compute the injection water as 51.108 — 1.208 = 49.9 lb. and the heat 
absorbed by it as approximately 49.9(73.4 — 50) = 1167 B. t. u. The 
1.208 lb. of feed were discharged at 73.4°, whereas the boiler feed was at 
59° ; a heat rejection of 73.4 - 59 = 14.4° occurred, or 14.4 x 1.208 = 17.4 
B. t. u. The total heat rejection was then 1167 + 1.7.4 = 1184.4 B. t. u., 
to which we must add 47 B. t. u. from the jackets, giving a total of 
1231.4 B. t. u. Adding this to the work done, we have 1231.4 + 225 = 
1456.4 B. t. u. accounted for of the total 1551 B. t. u. supplied; the 
discrepancy is over 6 per cent. 

When surface condensers are used, the temperatures of discharge of 
the condensed steam and the condenser water are different and the weight 



HIRN'S ANALYSIS 299 

of water is ascertained directly. In other respects the computation would 
be as given.* 

503. Statements of Efficiency. Engines are sometimes rated on the basis of 
fuel consumption. The duty is the number of foot-pounds of work done in the 
cylinder per 100 pounds of coal burned. The efficiency of the plant is the quotient 
of the heat converted into work per pound of coal, by the heat nnits contained in 
the pound of coal. In the test in Art. 498, the coal consumption per Ihp.-hr. 
was 2068.84 -=- 1338.62 = 1.54 ]b. In some cases, all statements are based on the 
brake liorse potoer instead of the indicated horse power. The ratio of the two is of 
course the mechanical efficiency. It may be noted that the engine is charged with 
steam, not at boiler pressure, but at the pressure in the steam pipe. The differ- 
ence between the two pressures and qualities represents a loss which may be con- 
sidered as dependent upon the transmissive efficiency. The plant efficiency is 
obviously the product of the efficiencies of boiler (Art. 574), transmission, and 
engine. 

504. Measurement of Heat Transfers : Hirn's Analysis. In the refined methods 
of studying steam engine performance developed by Hirn (41), and expounded by 
Dwelshanvers-Dery (42), the heat absorbed p 
and that rejected are both measured. Dur- 
ing any path of the cycle, the heat inter- 
change between fluid and walls is computed 
from the change in internal energy, the heat 
externally supplied or discharged, and the 
external work done. 

In Fig. 225, consider the cycle as made 
up of the four paths, 01, 12, 23, 30, called 
respectively a, b c, d Let ^ represent the 
weight ot cushion steam, and M that of 

cylinder feed, per stroke. We have then the following expressions for internal 
energy : ^ = M ^ + ^ . ^ =(M + M) (h 2 + x 2 r 2 ) ; 

E x =(M + M)(hx+xxrx) ; E 3 = ilf (/* 3 + x 3 r s ). 

The general equation for heat transfers is H = T + / + W, H standing for 
heat supplied or withdrawn, T + / for a change in internal energy, and W for 
external work done or consumed. In order to avoid confusion in algebraic signs, 
we will regard + H as representing a reception of heat by the fluid, + W as 
denoting positive ivork done by it, and + (T + J) (here represented by the symbol 
E with a subscript) as specifying a gain of internal energy. Let Q a , Q b , Q c , Q d , 
represent amounts of heat transferred to the walls along the paths a, b, c, d. 

Consider the path a. Let the heat supplied by the incoming steam be Q. 

TheU ' Q-Qa=W a +(Ex-E ). 

* It is most logical to charge the engine with the heat measured above the tem- 
perature of heat rejection. This, in Fig. 182, for example, makes the efficiency 

.-. J ,„ i rather than - ^_ c , the ordinate TX representing the feed-water temperature. 




300 



APPLIED THERMODYNAMICS 



Along the path b, - Q b = W b + (E 2 - E x ) ; along d, - Q d = - W d + (E - E$). 

Along c, heat is carried away by the discharged steam and by the cooling 
water. Let G denote the weight of cooling water per stroke, h 5 and h 4 its final and 
initial heat contents, and h G the heat contents of the discharged steam. The heat 
rejected by the fluid per stroke is then G(h- — h 4 ) + Mh 6 . Then — Q c — GQi- —h 4 ) 



Mir 



W c + (£3 - E 2 ), and Q c = - G(1 H - h 4 ) - Mh 6 + W c - (E z - E 2 ). 



The values of x at the four points of the cycle are obtained by comparing the 
volumes at those points with the volumes of saturated steam at the same pressure. 
If the cushion steam and cylinder feed per stroke, and the quality of the latter as 
supplied, be known, with the values of Ji 4 , h 5 , and h 6 and the weight of cooling 
water, we may then find values of Q a , Q b , Q n and Q d from the indicator diagram 
alone, the OP axis being properly located. 

505. Graphical Representation. In Fig. 226, from the base line xij, we may 
lay off the areas oefs representing heat lost during admission, smba showing heat 

gained during expansion, mhcr showing heat gained 
during release, and oakr showing heat lost during 
compression. If there were no radiation losses 
from the walls to the atmosphere, the areas above 
the line xy would just equal those below it. Any 
excess in upper areas represents radiation losses. 
Ignoring these losses, Hirn found by comparing the 
work done with the value of Q — J/7* 6 — G(h 5 — h 4 ) 
an approximate value for the mechanical equivalent 
of heat (Art, 32). 

Analytically, if Q r denote the loss by radiation, 
its value is the algebraic sum of Q a , Q b , Q c , Q d . 
If the heat Qj be supplied by a steam jacket, then 
Q r = Qj 4- 2Qa, i,c, d- The heat transfer during re- 
lease, Qc, regarded by Hirn as in a special .sense a measure of wastefulness of 
the walls, maybe expressed as Q r - Qj — 2Q a , 6, d- In a non-condensing engine, 
Q r can be determined only by direct experiment. In most 
cases the value of M is computed on the assumption that 
z 3 = 1.0 (Art. 440). 

Types of Steam Engine 

506. Special Engines. We need not consider the com- 
mercially unimportant class of engines using vapors other 
than steam, those experimental engines built for educa- 
tional institutions which belong to no special type (4 ; >), 
engines of novel and limited application like those em- 
ployed on motor cars (44), nor the " tireless " or stored hot- 
water steam engines occasionally employed for locomotion 
(45). 

A novel form of heat engine, the pulsometer, is shown 
in Fig. 227. It is employed solely for pumping water. 
Steam enters at B, water at E. The ball C being in the 





Fig. 227. Art. 506. 
Pulsometer. 



TYPES OF ENGINE 



301 



position shown, the steam forces water contained in A through the check valve V 
into a delivery passage D. When the water level sinks so far that steam begins to 
blow through D, violent agitation is produced, and the steam begins to condense. 
The partial vacuum causes the ball C to rock over so as to close chamber A, and 
also causes water to rise through E and the suction check valve X, again filling A. 
Meanwhile, the same series of actions has started in IF. The only moving parts 
are the ball and check valves. The efficiency is usually under 2 per cent. 
Letting the heat lost by the steam be x L + h — h v that gained by the water 
being h Y — h. 2 , the heat equation for y pounds of water pumped per pound of 



steam used is x ft L + h { 



y(h l — h 2 ). If the total head be s, the work is 



s(y + 1) foot-pounds (ignoring the fact that the condensed steam is received 
without head), and the efficiency is s(y + 1) -=- (x L + h — k^. 



507. Classification of Engines. Commercially important types may be con- 
densing or non-condensing. They are classified as right-hand or left-hand, accord- 
ing as the flywheel is on the right or left side of the center line of the cylinder, 
as viewed from the back cylinder head. They may be simple or multiple-expan- 




FlG. 228. Art. 507. — Angle-Compound Engine. (American Ball Engine Company.) 



302 



APPLIED THERMODYNAMICS 



sion, with all the successive stages and cylinder arrangements made possible in 
the latter case. They may be single-acting or double-acting ; the latter is the far 
more usual arrangement. They may be rotative or non-rotative. The direct-acting 
pumping engine is an example of the latter type ; the work done consists in a 
rectilinear impulse at the water cylinders. In the duplex engine, simple cylinders 
are used side by side. The terms horizontal, vertical, and inclined refer to the posi- 
tions of the center lines of the cylinders. The horizontal engine, as in Figs. 186 
and 229, is mostly used in land practice ; the vertical engine is most common at 




FlG. 229. Art. 507. — Automatic Engine. (American Ball Engine Company.) 



sea. Cross-compound vertical engines are often direct-connected to electric gen- 
erators. Vertical engines have occasionally been built with the cylinder below 
the shaft. This type, with the inclined engine, is now rarely used. Inclined 
engines have been built with oscillating cylinders, the use of a crosshead and 
connecting rod being avoided by mounting the cylinder on trunnions, through 
which the steam was admitted and exhausted. Figure 228 shows a section of 



TYPES OF ENGINE 



303 



the interesting angle-compound, in which a horizontal high-pressure cylinder 
exhausts into a vertical low-pressure cylinder. A different type of engine, but 
with a similar structural arrangement, has been used in some of the largest 
power stations. 

Engines are locomotive, stationary, or marine. The last belong in a class by 
themselves, and will not be illustrated here ; their capacity ranges up to that of 
our largest stationary power plants. Stationary engines are further classed as 
pumping engines, mill engines, power plant engines, etc. They may be further 
grouped according to the method of absorbing the power, as belted, direct-con- 
nected, rope drive, etc. An engine directly driving an air compressor is shown in 
Fig. 86. " Rolling mill engines " undergo enormous 
variations in load, and must have a correspondingly 
massive (tangye) frame. Power plant engines gen- 
erally must be subjected to heavy load variations ; 
their frames are accordingly usually either tangye or 
semi-tangye. Mill engines operate at steadier loads, 
and have frequently bee*i built with light girder 
frames. Modern high steam pressures have, however, 
led to the general discontinuance of this frame in 
favor of the semi-tangye. 

A slow-speed engine may run at any speed up to 
125 r. p. m. From 125 to 200 r. p. m. may be re- 
garded as medium speed. Speeds above 200 r. p. m. 
are regarded as high. Certain types of engine are 
adapted only for certain speed ranges ; the ordinary 
slide-valve engine, shown in Fig. 186, may be oper- 
ated at almost any speed. For large units, speeds 
range usually from 80 to 100 r. p.m. The higher- 
speed engines are considered mechanically less re- 
liable, and their valves do not lend themselves to quite 
as economical a distribution of steam. An important 
class of medium-speed engines has, however, been in- 
troduced, in which the independent valve action of 
the Corliss type has been retained, and the promptness 
of cut-off only attainable by a releasing gear has been 
approximated. In some cheap high-speed engines 
governing is effected simply but uneconomically by 
throttling the steam supply. Such engines may have 
shallow continuous frames or the sub-base, as in Fig. 
229, which represents the large class of automatic 
high-speed engines in which regulation is effected by 
automatically varying the point of cut-off. Figure 230 
shows three sets of indicator diagrams from a com- 
pound engine of this type, running non-condensing 
at various loads. Some of the irregulations of these 
diagrams are without doubt due to indicator inertia ; but they should be care- 
fully compared with those showing the steam distribution with a slow-speed 




304 



APPLIED THERMODYNAMICS 



releasing gear, in Fig. 218. All of the so-called " automatic " engines run at 
medium or high rotative speeds. 

The throttling engine is used only in special or unimportant applications. The 
automatic type is employed where the comparatively high speed is admissible, in 
units of moderate size. Better distribution is afforded by the four-valve engine, in 




,Corliss Steam Volve. [ 



x*r 



Steam pipe 



Bach Cylinder Head^ 
Boch Cylinder Head Studs 
Back CuJ Heod I 



Corliss Exhoust Valve 



r'Sfeom f lonqe 
_ 1 




.Throttle Volve 
Plomshed Steel Laqqinq 
titot insulatinq Fillinq 

Corliss SteomVolve Chamber 
ropt Cylinder Heod 

ont. Cylindtr Heod 5 tods 

n Rod 6 land Studs 
iston Rod Glond 

1 



■ stonRod Pocrum 



Exhoust Chest 



I ^^^ Exhaust Flonqe 
L-JiJ ^Exhoust Opemnq 

\f*l 



Corliss Exhoust Volve 
Planished Sheet Steel Loogina. 
eat insulating Filling 



houst Pipe 
Fig. 231. Art. 507. — Corliss Engine Details. (Murray Iron Works Company.) 



THE STEAM POWER PLANT 



305 



which the four events of the stroke may be independently adjusted, and this type 
is often used at moderately high speeds. Sharpness of cut-off is usually obtainable 
only with a releasing gear, in which the mechanism operating the valves is discon- 
nected, and the steam valve is au- 
tomatically and instantaneously 
closed. This feature distinguishes 
the Corliss type, most commonly 
used in high-grade mill and power 
plant service. With the releasing 
gear, usual speeds seldom exceed 
100 r. p. m. The valve in a Cor- 
liss engine is cylindrical, and ex- 
tends across the cylinder. Some 
details of the mechanism are 
shown in Fig. 231. In very large 
engines, the releasing principle is 
sometimes retained, but with 
poppet or other forms of valve. 
Figure 232 shows the parts of a 
typical Corliss engine with semi- 
tangye frame. 

508. The Steam Power Plant. 

Figure 233, from Heck (46), is 
introduced at this point to give 
a conception of the various ele- 
ments composing, with the en- 
gine, the complete steam plant. 
Fuel is burned on the grate 1; 
the gases from the fire follow 
the path denoted by the arrows, 
and pass the damper 4 to the 
chimney 5. Water enters, 
from the pump IV, the boiler 
through 29, and is evaporated, 
the steam passing through 8 to 
the engine. The exhaust steam 
from the engine goes through 
18 to the condenser III, to 
which water is brought through 
21. Steam to drive the condenser pump comes from 26. Its exhaust, 
with that of the feed pump 31, passes to the condenser through 27. The 
condensed steam and warmed water pass out through 23, and should, if 
possible, be used as a source of supply for the boiler feed. The free ex- 
haust pipe 19 is used in case of breakdown at the condenser. 




306 



APPLIED THERMODYNAMICS 




509. The Locomotive. 

This is an entire power plant, 
made portable. Figure 234 
shows a typical modern form. 
The engine consists of two 
horizontal double acting cyl- 
inders coupled to the ends of 
the same axle at right an- 
gles. These are located un- 
der the front end of the 
boiler, which is of the type 
described in Art. 563. A 
pair of heavy frames sup- 
ports the boiler, the load be- 
ing carried on the axles by 
means of an intervening 
" spring rigging." The stack 
is necessarily short, so that 
artificial draft is provided by 
means of an expanding noz- 
zle in the "smoke box," 
through which the exhaust 
steam passes; live steam 
may be used when necessary 
to supplement this. The 
engines are non-condensing, 
but superheating and heat- 
ing of feed water, particu- 
larly the former, are being 
introduced extensively. The 
water is carried, in an aux- 
iliary tender, excepting in 
light locomotives, in which a 
" saddle " tank may be built 
over the boiler. 

The ability of a locomo- 
tive to start a load depends 
upon the force which it can 
exert at the rim of the driv- 
ing wheel. If d is the cylin- 
der diameter in inches, L the 
stroke in feet, and p the 
maximum mean effective 
pressure of the steam per . 
square inch, the work done 
per revolution by two equal 
cylinders is wd 2 Lp. Assume 



THE LOCOMOTIVE 



307 



Trd^Lp -4- irD 



this work to be trans- 
mitted to the point of 
contact between wheel 
and rail without loss, 
and that the diameter 
of the wheel is D feet, 
then the tractive power, 
the force exerted at 
the rim of the wheel, 
d*Lp 
D 

The value of p, with 
such valve gears as are 
employed on locomo- 
tives, may be taken at 
80 to 85 per cent of the 
boilter pressure. The 
actual tractive power, 
and the pull on the 
drawbar, are reduced 
by the friction of the 
mechanism ; the latter 
from 5 to 15 per cent. 
Under ordinary con- 
ditions of rail, the 
wheels will slip when 
the tractive power ex- 
ceeds 0.22 to 0.25 the 
total weight carried by 
the driving wheels. 
This fraction of the 
total weight is called 
the adhesion, and it is 
useless to make the 
tractive power greater. 
In locomotives of cer- 
tain types, a " traction 
inereaser " is sometimes 
used. This is a device 
for shifting some of the 
weight of the machine 
from trailer wheels to 
driving wheels. The 
weight on the drivers 
and the adhesion are 
thereby increased. The 
engineman, upon ap- 




7, 9I /e* ,H" 



308 APPLIED THERMODYNAMICS 

proaching a heavy grade, may utilize a higher boiler pressure or a later cut-off 
than would otherwise be useful. 

510. Compounding. Mallet compounded the two cylinders as early as 1876. 
The steam pipe between the cylinders wound through the smoke box, thus becom- 
ing a reheating receiver. Mallet also proposed the use of a pair of tandem compound 
cylinders on each side. The Baldwin type of compound has two cylinders on each 
side, the high pressure being above the low pressure. Webb has used two ordinary 
outside cylinders as high-pressure elements, with a very large low-pressure cylinder 
placed under the boiler between the wheels. In the Cole compound, two outside 
low-pressure cylinders receive steam from two high-pressure inside cylinders. The 
former are connected to crank pins, as in ordinary practice : the latter drive a 
forward driving axle, involving the use of a crank axle. The four crank efforts 
differ in phase by 90°. This causes a very regular rotative impulse,, whence the 
name balanced compound. Inside cylinders, with crank axles, are almost exclusively 
used, even with simple engines, in Europe : two-cylinder compounds with both 
cylinders inside have been employed. The use of the crank axle has been complicated 
in some locomotives with a splitting of the connecting rod from the inside cylinders 
to cause it to clear the forward axle. Greater simplicity follows the standard 
method of coupling the inside cylinders to the forward axle. 

511. Locomotive Economy. The aim in locomotive design is not the greatest 
economy of steam, but the installation of the greatest possible power-producing 
capacity in a definitely limited space. Notwithstanding this, locomotives have 
shown Very fair efficiencies. This is largely due to the small excess air supply 
arising from the high rate of fuel consumption per square foot of grate (Art. 564). 
The locomotive's normal load is what would be considered, in stationary practice, 
an extreme overload. Its mechanical efficiency is therefore high. For the most 
complete data on locomotive trials, the Pennsylvania Railroad Report (47) should 
be consulted. The American Society of Mechanical Engineers has published a 
code (48) ; Reeve has worked out the heat interchange in a specimen test by Hirn's 
analysis (49). (See Art. 554.) 

(1) D. K. Clark, Bailway Machinery. (2) Isherwood, Experimental Besearches 
in Steam Engineering, 1863. (3) Be la condensation de la vapeur, etc., Ann. des 
mines, 1877. (4) Bull, de la Soc. Indust. de Mulhouse, 1855, et seq. (5) Broc. Inst. 
Civ. Eng., CXXXII. (6) Peabody, Thermodynamics, 1907, 233. (7) The Engineer- 
ing Magazine, December, 1906, 425. (8) Min. Broc. Inst. C. E., March, 1888 ; April, 
1893. (9) Op. cit. (10) Engine Tests, G. H. Barrus. (11) The Steam Engine, 
1892, p. 190. (12) The Steam Engine, 1905, 109, 119, 120. (13) Broc. Inst. Mech. 
Eng., 1889, 1892, 1895. (14) Ripper, Steam Engine Theory and Bractice, 1905, p. 167. 
(15) Ripper, op. cit., p. 149. (16) Trans. A. S. M. E., XXVIII, 10. (17) For a 
discussion of the interpretation of the Boulvin diagram, see Berry, The Temperature- 
Entropy Diagram, 1905. (18) Broc. Inst. Mech. Eng., January, 1895, p. 132. 
(19) The Steam Engine, 1906. (21) Trans. A. S. M. E., XV. (22) Ibid., XIII, 
647. (23) Ibid., XIX, 189. (24) Ibid., loc. cit. (25) Ibid., XXV, 482,483, 490, 492. 
(26) Manuel du Conducteur des Machines Binaires, Lyons, 1850-1851. (27) Pea- 
body, Thermodynamics, 1907, 283. (28) Thurston, Engine and Boiler Trials, p. 130. 
(29) Ripper, Steam Engine Theory and Bractice, 1905, p. 412. (30) Experimental 



THE STEAM ENGINE 309 

Engineering, 1907. (31) The Steam Engine Indicator, 1898. Reference should also 
be made to Miller's and Hall's chapters of Practical Instructions for using the Steam 
Engine Indicator, published by the Crosby Steam Gage and Valve Company, 1905. 
(32) Low, op. cit., pp. 103-107 ; Carpenter, op. cit., pp. 41-55, 531, 780. (33) Op. cit., 
p. 391. (34) Trans. A. S. M. E., VI, 716. (35) Ibid., XXV. (36) Ibid., 1892, also 
XXV, 827. (37) Ibid., XI. (38) Ibid., XXIV, 713. (39) Op. cit., 144. (40) The 
Steam Engine, p. 212. (41) Bull, de la Soc. Ind. de Mulhouse, 1873. (42) Expose 
Succinct, etc. ; Revue Universelle des Mines, 1880. (43) Carpenter, Experimental 
Engineering, 1907, 657 ; Peabody, Thermodynamics, 1907, 225. (44) Trans. A. S. 
31. E., XXVIII, 2, 225. (45) Zeuner, Technical Thermodynamics (Klein), II, 449. 
(46) The Steam Engine, 1905, I, 2, 3. (47) Locomotive Tests and Exhibits at the 
Louisiana Purchase Exposition, 1906. (48) Trans. A. S. M. E., 1892. (49) Ibid., 
XXVIII, 10, 1658. 

SYNOPSIS OF CHAPTER XIII 
Practical Modifications of the Bankine Cycle 

With valves moving instantaneously at the ends of the stroke, the engine would operate 
in the non-expansive cycle. The introduction of cut-off makes the cycle that of 
Bankine, modified as follows : — 

(1) Port friction reduces the pressure during admission, theoretically along a line of 
constant total heat. This dries or superheats the steam, but causes a loss of availa- 
bility of the heat. The piston speed influences the shape of the admission line. 
Regulation by throttling is wasteful. 

(2) The expansion curve differs in shape and position from that in the ideal cycle. 
Expansion is not adiabatic. The steam at the point of cut-off contains from 25 
to 70 per cent of water on account of initial condensation. Further condensation 
occurs early in the expansion stroke, followed by reevaporation later on, after 
the pressure has become sufficiently lowered. The inner surfaces only of the walls 
fluctuate in temperature. Condensation is influenced by 

(a) the temperature range : wide limits, theoretically desirable, introduce some 

practical losses ; 
(6) the size of the engine : the exposed surface is proportionately greater in 

small engines ; 

(c) its speed : high speed gives less time for heat transfers ; 

(d) the ratio of expansion : wide ratios increase condensation and decrease 
efficiency, particularly because of increased initial condensation. Initial 
wetness facilitates the formation of further moisture. In good design, the 
ratio should be fixed to obtain reasonably complete expansion without 

excessive condensation, say at 4 or 5 to 1. M= -^^/iZ. 

x/X~Vpe 

Steam jackets provide steam insulation at constant temperature ; they oppose initial 
condensation in the cylinder and are generally used with slow speeds and high 
ratios of expansion. Some saving is always shown. Superheat, used under similar 
conditions, increases the mean temperature of heat absorption. Each 75° of super- 
heat may increase the dryness at cut-off by 10 per cent. Superheat increases 
efficiency, and is preferable to increased initial pressure. The actual expansion 



310 : APPLIED THERMODYNAMICS 

curve, PV = pv, crosses the adiabatic. M. E. P. = p b(l + log e r) _ Fd with the 

r 

Kankine form of cycle. H.P. = 2 x diagram factor x m^L.Y 
J 33000 

(3) The exhaust line shows back pressure due to friction of ports, the presence of air, 
and reevaporation. High altitudes increase the capacity of non-condensing engines. 

(4) Clearance varies from 2 to 10 per cent. "Real" and "apparent" ratios of 
expansion. 

(5) Compression brings the piston to rest quietly ; though theoretically less desirable 
than jacketing, it may reduce initial condensation if properly limited. 

(6) Valve action is not instantaneous, and the corners of the diagram are always some- 
what rounded. 

The Steam Engine Cycle on the Entropy Diagram 

Cushion steam, present throughout the cycle, is not included in measurements of 

steam used. 
Its volumes may be deducted, giving a diagram representing the behavior of the 

cylinder feed alone. 
The indicator diagram shows actions neither cyclic nor reversible : it depicts a 

varying mass of steam. 
The Boulvin diagram gives the NT history correctly along the expansion curve 

only. 

The Beeve diagram eliminates the cushion steam ; it correctly depicts both expan- 
sion and compression curves, as referred to the cylinder feed. 

Diagrams may show (a) loss by condensation, (6) gains by increased pressure and 
decreased back pressure, (c) gains by superheating and jacketing. 

Multiple Expansion 

Increased initial pressure and decreased back pressure pay best with wide expansive 
ratios. 

Such ratios are possible, with multiple expansion, without excessive condensation. 

Condensation" is less serious because of (a) the use made of reevaporated steam, 
(&) the decrease in initial condensation, and (c) the small size of the -high- 
pressure cylinder. 

Several numbers and arrangements of cylinders are possible with expansion in two, 
three, or four stages. 

Incidental advantages : less steam lost in clearance space ; compression begins later ; 
the large cylinder is subjected to low pressure only ; more uniform speed and 
moderate strains are possible. 

The Woolf engine had no receiver ; the low-pressure cylinder received steam through- 
out the stroke as discharged by the high-pressure cylinder. The former, therefore, 
worked without expansion. The piston phases coincided or differed by 180°. 

In the receiver engine, the pistons may have any phase relation and the low-pressure 
cylinder works expansively. Early cut-off in the low-pressure cylinder increases 
its proportion of the load, and is practically without effect on the total work of the 
engine.* 



THE STEAM ENGINE ., 311 

The point of low-pressure cut-off to eliminate drop may be graphically or analytically 
determined for tandem and cross-compound engines. 

Tbe methods given ignore angularity of the connecting rod, clearance, and friction in 
passages ; they assume all expansive paths to be hyperbolic. 

In combining diagrams, two saturation curves are necessary, unless the cushion steam 
be deducted. 

The diagram factor has an approximate value the same as that in a simple engine hav- 
ing -tyn expansions, in which n is the number of expansions in the compound 
engine and c its number of expansive stages. 

Cylinder ratios are 3 or 4 to 1 if non-condensing, 4 or 6 to 1 if condensing, in com- 
pounds ; triples have ratios from 1:2.0:2.0 to 1:2.5:2.5. A large high-pressure 
cylinder gives high overload capacity. 

The engine may be designed by computing the m. e. p. of the combined ideal diagrams 
and dividing this between the cylinders so as to equalize work areas, or by assum- 
ing the cylinder ratio, the maximum practicable value of which is related to the 
total ratio of expansion. 

Governing should be by varying the point of cut-off in both cylinders. 

Drop in any but the last cylinder is usually considered undesirable. 

Exceptionally high efficiency is shown by compounds having cylinder ratios of 7 to 1. 
The high-pressure cylinder in ordinary compounds is too large for highest efficiency. 

The binary vapor engine employs the waste heat of the exhaust to evaporate a fluid 
having a lower boiling point than can be attained with steam. Additional work 
may then be evolved down to a rejection temperature of 60 or 70° E. The best 
result achieved is 167 B. t. u. per Ihp.-minute. 

Engine Tests 

The indicator measures pressures and volumes in the cylinder and thus shows the 

"cycle." 
Its diagram gives the m. e. p. and points out errors in valve adjustment or control. 

hi(w-\- W)-wh—Wh 



Calorimeters : the barrel type : Xq 



surface condensing : Xq = 



WU 

wh\ + Wh 2 — %oh — Who 
WL ~ 



superheating : xo = ^-= 1 limits of capacity ; 

itor«:ft=.:5±*<r-')-».-«. 



-to 

separating : direct weighing of the steam and water; 
chemical : insolubility of salts in dry steam ; 
electrical : 1 B. t. u. = 17.59 watts per minute. 

Engine trials : we may measure either the heat absorbed or the heat rejected + the work 

done. 
By measuring both, we obtain a heat balance. 
Results usually stated : lb. dry or actual steam per Ihp.-hr.; B. t. u. per Ihp.-minute ; 

thermal efficiency ; work per lb. Lteam ; Carnot efficiency ; Clausius efficiency ; 

efficiency ratios. 



312 APPLIED THERMODYNAMICS 

By assuming the steam dry at compression and release, and knowing the clearance, we 
may roughly estimate steam consumption from the indicator diagram. Reasonable 
accuracy is possible if the quality of steam at these points be known ; no informa- 
tion is then necessary other than that given by the diagrams themselves. 

Duty = ft.-lb. of work per 100 lb. coal. Plant efficiency = B - *• u - of work m Mechani- 

Brake hp. R *■ u ' in coal 

cat efficiency = *- — . 

Indicated hp. 

IHm's analysis: E x = 'LM (h x + x x r x ); H X = E X + W x ; heat transfer to and from 
walls may be computed from the supply of heat, the change in internal energy, 
and the work done. The excess of losses over gains represents radiation. 



Types of Steam Engine 

The pulsometer : efficiency = s(y 4- 1) -=- (xqLq -Mo — hi). 

Standard engines : non-condensing or condensing ; right-hand or left-hand ; simple 
or multiple expansion ; single-acting or double-acting ; rotative or non-rotative ; 
duplex or single ; horizontal, vertical, or inclined ; locomotive, stationary (pump- 
ing, mill, power plant), or marine; belted, direct-connected, or rope-drive; air 
compressors ; girder, tangye or semi-tangye frames ; slow, medium, or high speed ; 
throttling, automatic, four-valve, or releasing gear. 

The power plant : feed pump, boiler, engine, condenser. 

The locomotive : tractive power = - — £ ; adhesion = 0.22 to 0.25 x weight on drivers ; 

two-cylinder and four-cylinder compounds ; the balanced compound ; high econ- 
omy of locomotive engines. 

PROBLEMS 

1. Show from Art. 426 that the loss by a throttling process is equal to the prod- 
uct of the increase of entropy by the absolute temperature at the end of the process. 

2. Ignoring radiation, how fast are the walls gaining heat because of transfers 
during expansion in an engine running at 100 r. p. m., in which | pound of steam is 
condensed per revolution at a mean pressure of 100 lb., and 0.30 pound is reevaporated 
at a mean pressure of 42 lb ? 

3. Establish from Art. 434 an approximate formula for the relation between 
engine speed and wetness at cut-off in one of the tests. 

4. All other factors being the same, how much less initial condensation, at | cut- 
off, should be found in an engine 30£" x 48" than in one 7" x 7" ? 

5. Sketch a curve showing the variation in engine efficiency with ratio of expan- 
sion. 

6. Eind the percentage of initial condensation at i cut-off in an engine using dry 
steam, running at 100 r. p. m. with a pressure at cut-off of 120 lb., the engine being 
30i" x 48" (Art. 437). 

7. In Fig. 193, assuming the initial pressure to have been 100 lb., the feed-water 
temperature 90° F., find the approximate thermal efficiencies with the various amounts 
of superheat at a load of 15 hp. 



THE STEAM ENGINE 313 

8. In an ideal Clausius cycle with initially dry steam between p = 140 and p = 2 
(Art. 417), by what percentage would the efficiency be increased if the initial pressure 
were made 160 lb. ? By what percentage would it be decreased if the lower pressure 
were made 6 lb. ? 

9. Find the mean effective pressure in the ideal cycle with hyperbolic expansion 
and no clearance between pressure limits of 120 and 2 lb., with a ratio of expansion of 4. 

10. Find the probable indicated horse power of a double-acting engine with the 
best type of valve gear, jackets, etc., operating as in Problem 9, at 100 r. p.m., the 
cylinder being 30^" x 48". (Ignore the piston rod.) 

11. In Problem 9, what percentage of power is lost if the lower pressure is raised 
to 3| lb. ? 

12. By what percentage would the capacity of an engine be increased at an altitude 
of 10,000 ft. as compared with sea level, at 120 lb. initial pressure and a back pressure 
1 lb. greater than that of the atmosphere, the ratio of expansion being 4 ? (Atmos- 
pheric pressure decreases \ lb. per 1000 ft. of height.) 

13. An engine has an apparent ratio of expansion of 4, and a clearance amounting 
to 0.05 of the piston displacement. What is its real ratio of expansion ? 

14. In the dry steam Clausius cycle of Problem 8, by what percentages are the ca- 
pacity and efficiency affected if expansion Is hyperbolic instead of adiabatic ? Discuss 
the results. 

15. In the dry steam cycle of Problem 9, find the change in capacity and efficiency 
if the cycle is worked with hyperbolic compression to one fourth the initial pressure, 
clearance equal to 5 per cent of the piston displacement, hyperbolic expansion, 1 lb. of 
mean wiredrawing during admission, 70 per cent decrease in volume at cut-off due to 
initial condensation, and 2 lb. of mean extra back pressure during exhaust. 

16. In an engine having a clearance volume of 1.0 and a back pressure of 2 lb., 
the pressure at the end of compression is 40 lb. If the compression curve is PV 1 - 08 = c, 
what is the volume at the beginning of compression ? 

17. An engine works between 120 and 2 lb. pressure, the piston displacement 
being 20 cu. ft., clearance 5 per cent, and apparent ratio of expansion 4. The expan- 
sion curve is PV 1 - 02 = c, the compression curve PV 1 ' 08 = c, and the final compression 
pressure is 40 lb. Plot the PV diagram with actual volumes of the cushion steam 
eliminated. 

18. In Problem 16, 1.825 lb. of steam are present per cycle. Plot the entropy dia- 
gram from the indicator card by Boulvin's method. Plot the FiV diagram. 

19. In Problems 17 and 18, compute and plot the entropy diagram by Reeve's 
method, assuming the steam dry at the beginning of compression. (See Art. 394.) 
Discuss any differences between this diagram and that obtained in Problem 18. 

20. In a non-expansive cycle with dry steam at cut-off and no clearance, find the 
changes in capacity and economy by raising the initial pressure from 100 to 120 lb., 
the back pressure being 2 lb. 

21. A non-expansive engine with limiting volumes of 1 and 6 cu. ft. and an initial 
pressure of 120 lb., without compression, has its back pressure decreased from 4 to 2 lb. 
Find the changes in capacity and efficiency. The same steam is now allowed to expand 
hyperbolically to a volume of 21 cu. ft. Find the effects following the reduction of 
back pressure in this case. The steam is in each case dry at the point of cut-off. 



314 APPLIED THERMODYNAMICS 

22. Find the cylinder dimensions of an automatic engine to develop 30 horse 
power at 300 r. p. m., non-condensing, at \ cut-off, the initial pressure being 100 lb. 
and the piston speed 300 ft. per minute. The engine is double-acting. 

23. Sketch a possible cylinder arrangement for a quadruple-expansion engine with 
seven cylinders, three of which are vertical and four horizontal, showing the receivers 
and pipe connections. 

24. Using the ideal combined diagram for a compound engine with a constant 
receiver pressure, clearance being ignored, what must that receiver pressure be to 
divide the diagram area equally, the pressure limits being 120 and 2 and the ratio of 
expansion 16 ? 

25. Consider a simple engine 30|" x 48" and a compound engine 15|" and 
30^" x 48", all cylinders having 5 per cent of clearance and no compression. What 
are the amounts of steam theoretically wasted in filling clearance spaces in the simple 
engine and in the high-pressure cylinder of the compound, the pressures being as in 
Problem 24 ? 

26. Take the same engines. The simple engine has a real ratio of expansion of 4 ; 
the compound is as in Problems 24 and 25. Compression is to be carried to 40 lb. in 
the simple engine and to 60 lb. in the compound in order to prevent waste of steam. 
By what percentages are the work areas reduced in the two engines under consideration ? 

27. A cross-compound double-acting engine operates between pressure limits of 
120 and 2 lb. at 100 r. p. m. and 800 ft. piston speed, developing 1000 hp. Find the 
sizes of the cylinders under the following assumptions, there being no drop : (a) dia- 
gram factor 0.85, 20 expansions, receiver pressure 24 lb. ; (5) diagram factor 0.85, 
20 expansions, work equally divided ; (c) diagram factor 0.85, 20 expansions, cylinder 
ratio 5:1; (d) diagram factor 0.83, 32 expansions, work equally divided. Find the 
power developed by each cylinder in (a) and (c). Find the size of the cylinder of the 
equivalent simple engine having a diagram factor of 0.85 with 20 expansions. Draw up 
a tabular statement of the five designs and discuss their comparative merits. 

28. In Problem 27, Case (a), the receiver volume being equal to that of the high- 
pressure cylinder, find graphically and analytically the point of cut-off on the low- 
pressure cylinder. 

29 a. Find the point' of cut-off, as in Problem 28, if the engine is a tandem com- 
pound with 5 lb. of drop. 

29 b. In what respects are the results in Problems 27, 28, and 29 a to be modified 
so as to include the factors in Art. 473 and Art. 474 ? 

30. Trace the combined diagram for one end of the cylinder from the first set of 
cards in Fig. 230, assuming the clearance in each cylinder to have been 15 per cent of 
the piston displacement, the cylinder ratio 3 to 1, and the pressure scales of both cards 
to be the same. 

31. In Fig. 204 assume the steam to have been 70 per cent dry at cut-off, 95 per 
cent dry at the beginning of compression in the high-pressure cylinder, and 90 per 
cent dry at the beginning of compression in the low-pressure cylinder, the cylinder 
ratio being 4 ; and plot the combined diagram with cushion steam eliminated, showing 
the single saturation curve. 

32. Show on the entropy diagram the effect of reheating. 



THE STEAM ENGINE 315 

33. In Art. 483, what was the Carnot efficiency of the Josse engine ? Assuming 
it to have been used in combination with a gas engine, the maximum temperature in 
the latter being 3000° F,, by what approximate amount might the Carnot efficiency 
have been increased ? (The temperature of saturated sulphur dioxide at 35 lb. pres- 
sure is 52° F.) 

34. An indicator diagram has an area of 82,192.5 foot-pounds. What is the mean 
effective pressure if the engine is 30-£" x 48" ? What is the horse power of this engine 
if it runs double-acting at 100 r. p. m ? 

35. Given points 1, 2 on a hyperbolic curve, such that V-2—V\ = 15, Pi = 120, 
P 2 = 34.3, find the OP-axis. 

36. An engine develops 500 hp. at full load, and 62 hp. when merely rotating its 
wheel without external load. What is its mechanical efficiency ? 

37. Steam at 100 lb. pressure is mixed with water at 100°. The weight of water 
increases from 10 to 11 lb., and its temperature rises to 197J°. What was the percent- 
age of dryness of the steam ? 

38. The same steam is condensed in and discharged from a coil, its temperature 
becoming 210°, and 10 lb. of surrounding water rise in temperature from 100° to 2041°. 
Find the quality of the steam. What would have been an easier way of determining 
the quality ? 

39. What is the maximum percentage of wetness that can be measured in a throt- 
tling calorimeter in steam at 100 lb. pressure, if the discharge pressure is 30 lb. ? 

40. Steam at 100 lb. pressure has added to it from an external source 30 B. t. u. 
per pound. It is throttled to 30 lb. pressure, its temperature becoming 270.3°. What 
was its dryness ? 

41. In Problem 40, the added heat is from an electric current of 5 amperes pro- 
vided for one minute, the voltage falling from 220 to 110. What was the amount of 
heat added and the percentage of dryness of the steam ? 

42. An engine consumes 10,000 lb. of dry steam per hour, the moisture having 
been completely eliminated by a receiver separator which at the end of one hour is 
found to contain 285 lb. of water. What was the dryness of the steam entering the 
separator ? 

43. Check all results that can be checked in Arts. 498, 499, 500. 

A double-acting engine at 100 r. p.m. and a piston speed of 800 feet per minute 
gives an indicator diagram in which the pressure limits are 120 and 2 lb., the volume 
limits 1 and 21 cu ft. The apparent ratio of expansion is 4. The expansion curve 
follows the law PF 1,02 = c. Compression is to 40 lb., according to the law ?F 1,03 = c. 
Disregard rounded corners. The boiler pressure is 130 lb., the steam leaving the boiler is 
dry, the steam at the throttle being 95 per cent dry and at 120 lb. pressure. The boiler 
evaporates 26,500 lb. of steam per hour ; 2000 lb. of steam are supplied to the jackets 
at 120 lb. pressure. The engine runs jet-condensing, the inlet water weighing 530,000 
lb. per hour at 43.85° F., the outlet weighing 554,000 lb. at 90° F. The coal burned is 
2700 lb. per hour, its average heating value being 14,000 B. t. u. Compute as follows : 

44 a. The mean effective pressure and indicated horse power. (Note. The work 
quantities under the curves must be computed with much accuracy.) 

44 b. The cylinder dimensions of the engine. 



316 APPLIED THERMODYNAMICS 

45. The heat supplied at the throttle per pound of cylinder and jacket steam, and 
the B. t. u. consumed per Ihp. per minute ; the engine being charged with heat above 
the temperature of the condenser discharge (Art. 502). 

46. The dry steam consumption per Ihp.-hr., thermal efficiency, and work per 
pound of dry steam. 

47. The Carnot efficiency, the Clausius efficiency, and the efficiency ratio, taking 
the limiting conditions as at the throttle and the condenser outlet. 

48. The cylinder feed steam consumption computed as in Art. 500 ; the consump- 
tion thus computed but assuming x = 0.80 at release, x = 1.00 at compression. Com- 
pare with Problem 46. 

49. The steam consumption computed as in Art. 501 ; develop the expression 

3,9(30,000 (WB- VT) (1 + a) 
141 (P + p) 
for indicated steam consumption in a simple engine giving the same diagram at both 
ends of the cylinder and having the same clearance at each end. 

50. The percentage of steam lost by leakage (all leakage occurring between the 
boiler and the engine) ; the transmissive efficiency ; the unaccounted-for losses. 

51. The duty, the efficiency of the plant, and the boiler efficiency. 

52. The heat transfers and the loss of heat by radiation, as in Art. 504, assuming 
x = 1.00 at compression. Compare the unaccounted-for heat with that obtained in 
Problem 50. 

53. The value of the mechanical equivalent of heat which might be computed from 
the experiment. 

54. A pulsometer receives water at its own level and lifts it 30 feet. The dis- 
charge being at a temperature of 190°, and 0.001 lb. of dry steam being supplied per 
pound of water lifted, at 100 lb. pressure, find the efficiency. 

55. Explain the meaning of the figure 2068.84 in Art. 503. 

56. Revise Fig. 233, showing the arrangement of machinery and piping if a surface 
condenser is used. 

57. A locomotive weighing 200,000 lb. carries, normally, 60 per cent of its weight 
on its drivers. The cylinders are 19" x 26", the wheels C)6" in diameter. What is 
the maximum boiler pressure that can be profitably utilized ? If the engine has a trac- 
tion increaser that may put 12,000 lb. additional weight on the drivers, what maximum 
boiler pressure may then be utilized ? 

58. What is the percentage of error in the calculation of Art. 500 ? 

59. Represent Fig. 217 on the PV diagram. 




CHAPTER XIV 

THE STEAM TURBINE 

512. The Turbine Principle. Figure 235 shows the method of using steam in 
a typical impulse turbine. The expanding nozzles discharge a jet of steam at high 
velocity and low pressure against 
the blades or buckets, the im- 
pulse of the steam causing ro- 
tation. We have here, not 
expansion of high pressure steam 
against a piston, as in the ordi- 
nary engine, but utilization of 
the kinetic energy of a rapidly 
flowing stream to produce move- 
ment. One of the assumptions 
of Art. 11 can now no longer 
hold. All of the expansion oc- 
curs in the nozzle ; the expansion 
produces velocity, the velocity does FlG - 235 ' ^ wSefLTozTle? LaTal Turbine 
work. The lower the pressure 

at which the steam leaves the nozzle, the greater is the velocity attained. It will 
presently be shown that to fully utilize the energy of velocity, the buckets must 
themselves move at a speed proportionate to that of the steam. This involves ex- 
tremely high rotative speeds. 

The steps in the design of an impulse turbine are (a) determination 
of the velocity produced by expansion, (6) computation of the nozzle 
dimensions necessary to give the desired expansion, and (c) the propor- 
tioning of the buckets. 

513. Expansive Path. There is a gradual fall of pressure while the 
steam passes through the nozzle. With a given initial pressure, the pres- 
sure and temperature at any stated point along the nozzle should never 
change. There is, therefore, no tendency toward a transfer of heat be- 
tween steam and w r alls. Further, the extreme rapidity of the movement 
gives no time for such transfer ; so that the process in the nozzle is truly 
adiabatic, although friction renders it non-isentropic. The first problem 
of turbine design is then to determine the changes of velocity, volume, 
temperature or dryness, and pressure, during such adiabatic expansion, 
for a vapor initially wet, dry, or superheated ; the method may be accu- 

317 



318 APPLIED THERMODYNAMICS 

rate, approximate (exponential), or graphical. The results obtained are 
to include the effect of nozzle friction. 

514. The Turbine Cycle. Taking expansion in the turbine as adiabatic 
and as carried down to the condenser pressure, the cycle is that of Clausius, 
and is theoretically more efficient than that of any ordinary steam engine 
working through the same range. The turbine is free from losses due to 
interchange of heat ivith the walls. The practical losses are four : 

(a) Friction in the nozzles, causing a fall of temperature without the 
performance of work ; 

(6) Incomplete utilization of the kinetic energy by reason of the 
assumed blade angles and residual velocity of the emerging jet (Art. 528); 

(c) Friction along the buckets, increasing as some power of the stream 
speed ; 

(d) Mechanical friction of journals and gearing, and friction between 
steam and rotor as a whole. 

515. Heat Loss and Velocity. In Fig. 236, let a fluid flow adiabatically 
from the vessel a through the frictionless orifice b. Let the internal en- 
ergy of the substance be e in a and E in b ; the 
velocities v and V', the pressures p and P; and 
the specific volumes w and W. If the velocities 
could be ignored, as in previous computations, 
the volume of each pound of fluid in a would 
decrease by w in passing out at the constant 
pressure^; and the volume of each pound of 

Fig. 2H6. Art. 515. — Flow fi^id. in b would increase by W at the constant 

pressure P. The net external work done would 
be PW—pw, the net loss of internal energy e — E, and these two quan- 
tities would be equal. With appreciable velocity effects, we must also 
consider the kinetic energies in a and b ; these are 

£ and Yl, 
2y 2g' 

and we now have 

H=T+I+W+V, 

(T + T)+W+V=0, 

V 2 v 2 

E + PW+i- = e+pw + -£-, 

or X!-.vL=p W -PW+e-E. 

29 2g 




HEAT DROP AND VELOCITY 



319 



Let X, U, H, R, and x, u, h, r, be the dryness, increase of vol- 
ume during vaporization, heat of liquid, and internal latent heat, at 
PW and pw respectively ; let s be the specific volume of water; then 
for expansion of a vapor from pw to P W within the saturated region, 



— -=p(xu + s} 

2g '±g 



H-XR 



P(XU+ s)+h + xr 

= q -Q + s(p-py, 

in which q, Q represent total heats of wet vapor above 32 degrees. 
If expansion proceeds from the superheated to the saturated region, 
F 2 



2<7 2g 



= p](u + s) + (w-?i)\-P(XU+s) + h + r 



+ 



p(iv — n) 



H-XR, 



in which n = u + s is the volume of saturated steam at the pressure p, 
w is the volume of superheated steam, and 

p(w — n) 

y-i 

is the internal energy measured above saturation.* This also re- 
duces to q — Q + «(j? — P), where q is the total heat in the super- 
heated steam, and the same form of 
expression will be found to apply to 
expansion wholly in the superheated 
region. The gain in kinetic energy 
of a jet due to adiabatic expansion to 
a lower pressure is thus equivalent to 
the decrease in the total heat of the 
steam plus the work which would be 
required to force the liquid back 
against the same pressure head. In 
Fig. 237, let ah,. AB, CD, represent the three paths. Then the 
losses of heat are represented by the areas dabc, deABc, deCDfc. 

* For any gas treated as perfect, the gain of internal energy from t to T is 




Fig. 237. 



Art. 515. — Adiabatic Heat 
Drop. 



l(T-t) 



y 







(r-o = 



PV — pv 



y - i y — i 

or in this case, since internal energy is gained at constant pressure, 

_ p{w — n) 

y-l 



320 APPLIED THERMODYNAMICS 

The term s(p — P) being ordinarily negligible, these areas also rep- 
resent the kinetic energy acquired, which may be written, 

V 2 v 2 n 

ST" 5 *"*"* 

In the turbine nozzle, the initial velocity may also, without serious 
error, be regarded as negligible ; whence 

V 2 

2^ = q- Q or V= V50103.2 (q- Q) = 223.84 Vq-Q feet per second. 

516. Computation of Heat Drop. The value of q — Q may be determined 
for an adiabatic path between stated limits from the entropy diagram, 
Fig. 175, or from the Mollier diagram, Fig. 177. Thus, from the last 
named, steam at 100 lb. absolute pressure and at 500° F. contains 1273 
B. t. u. per pound ; steam 85 per cent dry at 3 lb. absolute pressure 
contains 973 B. t. u. Steam at 150 lb. absolute pressure and 600° F. con- 
tains 1317 B. t. u. If it expand adiabatically to 2.5 lb. absolute pressure, 
its condition becomes 88 per cent dry, its heat contents 1000 B. t. u., and 
the velocity produced is 

223.84 V3TT = 4000 ft. per second. 

517- Vacuum and Superheat. The entropy diagram indicates the nota- 
ble gain due to high vacua and superheat. Comparing dry steam expanded 
from 150 lb. to 4 lb. absolute pressure with the same steam superheated 
to 600° and expanded to 2.5 lbs. absolute pressure, we find q — Q in the 
former case to be 63 B. t. u., and in the latter, 317 B. t. u. The corre- 
sponding values of Fare 1770 and 4000 ft. per second. The turbine is 
peculiarly adapted to realize the advantages of wide ratios of expansion. 
These do not lead to an abnormally large cylinder, as in ordinary engines ; 
the " toe " of the Clausius diagram, Fig. 184, is gained by allowing the 
steam to leave the nozzle at the condenser pressure. Superheat, also, is 
not utilized merely in overcoming cylinder condensation; it increases the 
available " fall " of heat, practically without diminution. 

518. Effect of Friction. If the steam emerging from the nozzle were brought 
back to rest in a closed chamber, the kinetic energy would be reconverted into 
heat, as in a wiredrawing process, and the expanded steam would become super- 
heated. Watkinson has, in fact, suggested this (1) as a method of superheating 
steam, the water being mechanically removed at the end of expansion, before re- 
conversion to heat began. In the nozzle, in practice, the friction of the steam 
against the walls does partially convert the velocity energy back to heat, and the 
heat drop and velocity are both less than in the ideal case. 



EFFECT OF NOZZLE FRICTION 



321 



In Fig. 238, for adiabatic expansion from p, v, q, to P, V, Q, the 
velocity imparted is 



223.84 Vg - Q. 



During expansion from p, v, q, to P 1} Vj, Q x , 

the velocity imparted is 



223.84 Wq-Qi. 

Since V\ exceeds V, the steam is more nearly 
dry at V x ', i.e. Q x exceeds Q. The los's of 
energy due to the path pvq — PiViQi as 
compared with pcq — PVQ, is 



,P,v,q, 




P.V.Q, 



P, V Q 






Qi-Q, 



Fig. 238. Art. 518. — Abiabatic 
Expansion with and without 
Friction. 



in which X 2 is the difference of the squares of the velocities at Q and Q x . 

This gives X 2 = 50103.2 (Q x - Q). In Fig. 239, let NA be the adiabatic 

path, NX the modified path due to fric- 
tion. NZ represents a curve of constant 
total heat ; along this, no work would be 
done, but the heat would steadily lose its 
availability. As NX recedes from NA 
toward NZ, the work done during expan- 
sion decreases. Along NA, all of the heat 
lost (area FHNA) is transformed into 

' N work; along NZ, no heat is lost and no 
239. Art. 518. — Expansive '. _ e ' -n-nrTum 

Path as Modified by Friction. work 1S done > the areas ^FHNC and 

BFZD being equal. Along NX, the heat 
transformed into work is BFHNC - BFXE = FHNA - CAXE, less 
than that during adiabatic expansion by the amount of work converted 
back to heat. Considering expansion from N to Z, 




Fig. 



V= 223.84 Vg- Qi=0, 

since q = Q v Nozzle friction decreases the heat drop, the final velocity 
attained, and the external work done. 



519. Allowance for Friction Loss. For the present, we will assume 
nozzle friction to reduce the heat drop by 10 per cent. In Fig. 240, which 
is an enlarged view of a portion of Fig. 177, let AB represent adiabatic 
(isentropic) expansion from the condition A to the state B. Lay off 



BC = 



AB 

10 ' 



322 



APPLIED THERMODYNAMICS 



and draw the line of constant heat CD. 




Fig. 240. 

and 

If m = 



Arts. 519, 524, 525, 532, 534. 
of the Turbine. 



The Steam Path 



HG = 



Then D is the equivalent final 
state at the same pressure 
as that existing at B, and 
AC represents the heat 
drop corrected for friction. 
Similarly by laying off 
AH 
10 
and drawing GE to inter- 
sect the 35-lb. pressure 
line, we find the point E 
on the path AD of the 
steam through the nozzle. 
We may use the new heat 
drop thus obtained in de- 
termining V; or generally, 



is the friction loss, 



Z? 

2g 



(l-m)(q-Q) 



V= 223.84 Vl - m Wq- Q. 



0.10, V= 212.42 Vq-Q. 



520. Analytical Relations. The influence of friction in determining the final 
condition of the steam may be examined analytically. For example, let the initial 
condition be wet or dry ; then friction will not ordinarily cause superheating, so 
that the steam will remain saturated throughout expansion. Without friction, the 
final dryness x would be given by the equation (Art. 392), 

xl 

7 



iog* ™ 



x l 



T ' t ~ T 
Friction causes a return to the steam of the quantity of heat m(q — Q). This in- 



creases the final dryness by 



X n 



m(q - Q) 



making it 

xn 






Q) 



If the initial condition is superheated to t s , and the final condition saturated, 
adiabatic expansion would give 

fclogey 



t I 



XqIq 



and friction would make the final condition 



T^loge^+i+klogJjl+mCq 



Q) 



NOZZLE PROPORTIONS 



323 



If the steam is superheated throughout expansion, we have for the final tem- 
perature T s , without friction, 



l0g e ^ + - + £log/* = ^+£ l°gey 



in which the value of k must be obtained by successive approximations. 



521. Rate of Flow. 



For a flow of G pounds per second at the velocity V, when 

GW 
V ' 



the specific volume is W, the necessary cross-sectional area of nozzle is F 
The values of W and V may be 
read or inferred from the heat 
chart or the formulas just given. 
In Fig. 241 (2), let ah represent 
frictionless adiabatic expansion 
on the TN plane, a'b' the same 
process on the PV plane. By 
finding q a and values of Q at 
various points along ab, we may 
obtain a series of successive 
values of V. The correspond- 
ing values of W being read from 
a chart or computed, we plot the 
curve MN, representing the re- 
lation of specific volume and 
velocity throughout the expan- 
sion. Draw yy' parallel to W, 
making Oy = G, to some con- 
venient scale. Draw any line OD from to MN, intersecting yy' at k. 

similar triangles 




Art. 521. — Graphical Determination of 
Nozzle Area. 



From 



:yO::On: nD, or yk = ^ =F. 



To find the pressure at any specified point on the nozzle, lay off yk = F, draw 
OkD, Dn, and project z to the PT plane. The minimum value of F is reached 
when OD is tangent to MN. It becomes infinite when V = 0. The conclusion 
that the cross-sectional area of the nozzle reaches a minimum at a certain stage in the 
expansion will be presently verified. 



522. Maximum Flow. For a perfect gas, 

,E = 



PW 



If the initial velocity be negligible, we have, as the equation of flow (Art. 515), 



2^7 
and since 



pi -PW+-^-- 



PW 



y - 1 y - 1 y - 1 



y-(pw-PW); 



pivv = PW», PW=pw(—Y =pw(-\ V 



324 APPLIED THERMODYNAMICS 

Then 

From Art. 521, 



Taking the value of V at 



we obtain 



J 



^"(>-m 



>4»A*'[Hfn 



y+l 



G = _V ^^J \pj_ \ = Fp^2ff 






This reaches a maximum, for air, when P + p = 0.5214 (3). The velocity is then 
equal to that of sound. For dry steam, on the assumption that y = 1.135, and 
that the above relations apply, the ratio for maximum flow is 0.577. 

Using the value just given for the ratio P + p, with y = 1.402, the equation 
for G simplifies to 

G = 0.491 FpJ^f, 

the equation of flow of a permanent gas, which has been closely confirmed by 
experiment. With steam, the ratio of the specific heats is more variable, and the 
ratio of pressures has not been as well confirmed experimentally. Close approxi- 
mations have been made. Clarke (4), for example, shows maximum flow with 
saturated steam to occur at an average ratio of 0.56. The pressure of maximum 
flow determines the minimum or throat diameter of the nozzle, which is independ- 
ent of the discharge pressure. The emerging velocity may be greater than that 
in the throat if the steam is allowed to further expand after passing the throat. 
The nozzle should in all cases continue beyond the throat, either straight or ex- 
panding, if the kinetic energy is all to be utilized in the direction of flow. 

523. Experiments. Many experiments have been made on the flow of fluids 
through nozzles and orifices. Those of Jones and Rathbone (5), Rosenhain (6), 
Gutermuth (7), Napier (8), Rateau (9), Hall (10), Wilson (11), Kunhardt (12), 
Buchner (13), Kneass (14), Lewicki (15), Durley (16), and chiefly, perhaps, those 
of Stodola (17), should be studied. There is room for further advance in our 
knowledge of the friction losses in nozzles of various proportions. There are sev- 
eral methods of experimentation: the steam, after passing the orifice, may be con- 
densed and weighed; the pressure at various points in the nozzle may be measured 
by side orifices or by a searching tube ; or the reaction or the impulse of the steam 
at its escape may be measured. The velocity cannot be measured directly. 



TYPES OF TURBINE 



325 



^^ 



Fig. 242. Art. 
523.— Diverg- 
ing Orifice. 



A greater rate of flow is obtainable through an orifice in a thin plate (Fig. 
242) than through an expanding nozzle (Fig. 243). For pressures under 80 lb., 
with discharge into the atmosphere, the plain orifice is more efficient 
in producing velocity. For wider pressure ranges, a divergent 
nozzle is necessary to avoid deferred expansion occurring after 
emergence. Expansion should not, however, be carried to a pres- 
sure lower than that of discharge. The rate of flow, but not the 
emerging velocity, depends upon the shape of the inlet ; a slightly 
rounded edge (Fig. 243) gives the greatest rate; a greater amount 
of rounding may be less desirable. The experimentally observed 

critical pressure ratio { — , Art. 522) ranges with various fluids 

from 0.50 to 0.85. Maximum flow occurs at the lower ratios with rather sharp 
corners at the entrance, and at the higher ratios when a long divergence occurs 
beyond the throat, as in Fig. 243. The "most efficient" 
nozzle will have different proportions for different pressure 
ranges. The pressure is, in general, greater at all points 
along the nozzle than theory would indicate, on account of 
Fig. 243. Arts. 523, faction ; the excess is at first slight, but increases more and 
_ .— rpaucling more rapidly during the passage. Most experiments have 
necessarily been made on very small orifices, discharging to 
the atmosphere. The friction losses in larger orifices are probably less. The 
experimental method should include at least two of the measurements above 
mentioned, these checking each other. The theory of the action in the nozzle 
has been presented by Heck (18). Zeuner (19) has discussed the flow of gases to 
and from the atmosphere (20), both under adiabatic and actual conditions, and 
the efflux of gases in general through orifices and long pipes. 





<««« :<.<< 



BUCKET WHEEL 



524. Types of Turbine. The single stage impulse turbine of Fig. 
235 is that of De Laval. Its action is illustrated in Fig. 244. The 
pressure falls in the nozzle, and remains 

■*■ PRESSURES 

constant in the buckets. The Curtis and 
„««,„„„ Rateau turbines 
use a series of 
wheels, with ex- 
panding nozzles 
between the va- 
rious series (Figs. 

245, 246). The steam is only partially ex- 
Art. 524.— Curtis panded in each nozzle, until it reaches the 
last one. Such turbines are of the multi- 
stage impulse type. During passage through the blades, the ve- 
locity decreases, while the pressure remains unchanged. In the 



Fig. 244. 



f 



Art. 524. 
Turbine. 



De Laval 



Fig. 245. 



326 



APPLIED THERMODYNAMICS 




Z 



? 



pressure turbine of Parsons, there are no expanding nozzles ; the 
steam passes successively through the stationary guide vanes (7, g, 

and movable wheel buckets, IT, w, Fig. 247. 
A gradual fall of pressure occurs, the buck- 
teIm of ets being at all times full of steam. In 
impulse turbines, the buckets need not be 
full of steam, and the pressure drop occurs 
Kateau J n the nozzle only. 

A lower rotative speed results from the 
use of several pressure stages with expanding nozzles. Let the 
total heat drop of 317 B. t. u., in Art. 
516, be divided into three stages by three 
sets of nozzles. The exit velocity from 
each nozzle, corrected for friction, is 



Fig. 246. 



Art. 524. 
Turbine. 






aaacaa 



»>»)»*>•% 



(cua«ca 




Fig. 247. 



Arts. 524, 533. — Parsons 
Turbine. 



then 212.42 V?- Q = 2180 ft. per sec- 
ond, instead of 3790 ft. per second ; lay- 
ing off in Fig. 240 the three equal heat 
drops, we find that the nozzles expand between 150 and 50, 50 and 
13, and 13 and 2.5 lb. respectively. The rotative speeds of the 
wheels (proportional to the emerging velocities), Art. 528, are thus 
reduced. 

525. Nozzle Proportions ; Volumes. The specific volume W of the 
steam at any point along the path AD, Fig. 240, having been obtained 
from inspection of the entropy chart, or from the equation of condition, 
and the velocity F v at the same point having been computed from the 

wc 

heat drop, the cross-sectional area of the nozzle, in square feet, is F= ■ 

(Art. 521). Finding values of F for various points along the expansive 
path, we may plot the nozzle as in Fig. 243, making the horizontal inter- 
vals, ab, be, eel, etc., such that the angle between the diverging sides is 
about 10°, following standard practice. It has been shown that F reaches 
a minimum value when the pressure is about 0.57 of the initial pres- 
sure, and then increases as the pressure falls further. If the lowest 
pressure exceeds 0.57 of the initial pressure, the nozzle converges toward 
the outlet. Otherwise, the nozzle converges and afterwards expands, as 
in Fig. 243. Let, in such case, o be the minimum diameter, O the outlet 
diameter, L the length between these diameters ; then for an angle of 

10° between the sides, — = L tan 5°, or L = 5.715(0 — o). 



VELOCITY DIAGRAMS 



327 



526. Work Done. The work done in the ideal cycle per pound 
of steam is 778(^ — Q) foot-pounds. Since 1 horse power = 1,980,000 
foot-pounds per hour, the steam consumption per hp.-hr. is theoreti- 
cally 1,980,000 - 778(2 ~ Q) = 2545 -*-(?- 0- If E is tlie effi- 
ciency ratio of the turbine, from steam to buckets, and e the 
efficiency from steam to shaft, then the actual steam consumption 
per indicated horse power is 2545 -r- U(q — §), and per brake horse 
power is 2545 -r- e(q — Q~) pounds. The modifying influences of nozzle 
and bucket friction in determining E are still to be considered. 




Fig. 248. Art. 527. — Velocity Diagram. 



527. Relative Velocities. In Fig. 248, let a jet of steam strike 
the bucket A at the velocity v, the bucket itself moving at the speed 
u. The velocity of the steam rela- 
tive to the bucket is then repre- 
sented in magnitude and direction 
by V. The angles a and e made 
with the plane of rotation of the 
bucket wheel are called the absolute 
entering and relative entering angles 
respectively. Analytically, sin e = v 
sin a -7- V. The stream traverses 
the surface of the bucket, leaving it with the relative velocity x\ 
which for convenience is drawn as x from the point O. Without 

bucket friction, x- = V. The 
angle / is the relative angle of 
exit. Laying off u, from g,. we 
find Y as the absolute exit ve- 
locity, with g as the absolute 
angle of exit. Then, if x = V, 
sing= Vsinf-r- Y. 

To include the effect of nozzle 
and bucket friction, we proceed 
as in Fig. 249, decreasing v to 
Vl — m of its original value 
(Art. 519), and making x less than V by from 5 to 20 per cent, as 
in ordinary practice. As before, sin e = v sin a-±V\ but for a bucket 
friction of 10 per cent, sin# = 0.9 Fsin/-r- Y. 




Fig. 249. Arts. 527, 532, 534. — Velocity 
Corrected for Friction. 



328 



APPLIED THERMODYNAMICS 




Fig. 250. Arts. 528,529.— 
Rotative and Thrust 
Components. 



528. Bucket Angles and Work Done. In Fig. 250, the absolute 
velocities v and Y may be resolved into components ab and db in the 
direction of rotation, and ac and de at right 
angles to this direction. The former compo- 
nents are those which move the wheel ; the lat- 
ter produce an end thrust on the shaft. Now 
ab -+- bd (bd being negative) is the change in 
velocity of the fluid in the direction of rotation ; 
it is the acceleration ; the force exerted per 
pound is then 

(ab + bd) +g = (ab + bd) -f- 32.2 

= (vcos a 4- Ycosg) -r- 32.2. 

This force is exerted through the distance u 
feet per second ; the work done per pound of steam is then 
u(y cos a + Fcos^)-j- 32.2 foot-pounds. This, from Art. 526, equals 
778 E(q- Q) whence 

E=u(v cos a + Ycosg) -v- 25051.6(0 - <?). 

The efficiency is thus directly related to the bucket angles. 

To avoid splashing, the entrance angle of the bucket is usually 
made equal to the relative entering angle of the jet, as in Fig. 251. 
(These formulas hold only when the sides of the 
buckets are enclosed to prevent the lateral 
spreading of the stream.) In actual turbines, 
bd (Fig. 250) is often not negative, on account 
of the extreme reversal of direction that would 
be necessary. With positive values of bd, the 
maximum work is obtained as its value ap- 
proaches zero, and ultimately it is uv cos a -t- 32.2. 

v 2 
Since the kinetic energy of the jet is — , the Fig. 251. Art. 528.— 

& g Velocities and Bucket 

efficiency E from steam to buckets then becomes An s les - • 

In designing, we may either select an exit bucket angle 




ctU 

I - cos a 



which shall make bd equal to zero (the relative exit velocity being 
tangential to the surface of the bucket), or we may choose such an 
angle that the end thrust components de and ca, Fig. 250, shall bal- 



VELOCITY EFFICIENCY 



329 



ance. In marine service, some end thrust is advantageous ; in 
stationary work, an effort is made to eliminate it. This would be 
accomplished by making the entrance and exit bucket angles equal, 
for a zero retardation by friction. With friction considered, the 
angle of exit K, in Fig. 251, must be greater than the entering an- 
gle e. In any case, where end thrust is to be eliminated, the rota- 
tive component of the absolute exit velocity must be so adjusted as 
to have a detrimental effect on the economy. 

529. Effect of Stream Direction on Efficiency. Let the stream strike 
the bucket in the direction of rotation, so that the angle a — 0, Fig. 250, 
the relative exit velocity being perpendicular 
to the plane of the wheel. The work done is 

u ~ , while the kinetic energy is The 

9 ' _ ' 2 9 

efficiency, 2u — — 3 becomes a maximum at 

0.50 when u = - • With a cup-shaped vane, as 



in the Pelton wheel, Fig. 252, complete reversal -# _ 

of the jet occurs; the absolute exit velocity, 

ignoring friction, is v — 2u. The change in FlG - 252 - Arts. 529, 536.- 

■,•*.• o A, x -.,-.-, ton Bucket, 

velocity is v -\- v — Zu = 2{v — u), and the work 

is 2u(v— u) -~ g, whence the efficiency, — ^ ~~ u , becomes a maximum 

fl- 
at 100 per cent when u = -• Complete reversal in turbine buckets is im- 
practicable. 




530. Single-Stage Impulse Turbine. The absolute velocity of steam enter- 
the buckets is computed from the heat drop and nozzle friction losses. In a 

turbine of this type, the speed of the 
buckets can scarcely be made equal 
to half that of the steam ; a more 
usual proportion is 0.3. The velocity 
u thus seldom exceeds 1400 ft. per 
second. Fixing the bucket speed and 
the absolute entering angle of the 
steam (usually 20°) we determine 
graphically the entering angle of the 
bucket. The bucket may now be de- 
signed with equal angles, which would 
-*-t-^>\ eliminate end thrust if there were no 

Fig. 253. Art. 530. — Bucket Outline. friction, or, allowance being made for 




330 



APPLIED THERMODYNAMICS 



friction, either end thrust or the rotative component of the* absolute exit velocity 
may be eliminated. The normals to the tangents at the edges of the buckets being 

drawn, as ec, Fig. 253, 
the radius r is made 
equal to about 0.965 ec. 
The thickness t may 
be made equal to 0.2 
times the width kl. 
The bucket as thus 
drawn is to a scale as 
yet undetermined; 
the widths kl vary in 
practice from 0.2 to 
1.0 inch. 

It should be noted 
that the back, rather 
than the front, of the 
bucket is made tan- 
gent to the relative 
velocity V. The work 
per pound of steam 
being computed from 
the velocity diagram, 
and the steam con- 
sumption estimated 
for the assumed out- 
put, we are now in a 
position to design the 
nozzle. 

531. Multi-stage 
Impulse Turbine. If 
the number of pres- 
sure stages is few, as 
in the Curtis type, the 
heat drop may be di- 
vided equally between 
the stages. In the 
Kateau type, with a 
large number of 
stages, a proportion- 
ately greater heat drop 
occurs in the low-pres- 
sure stages. The cor- 
responding intermedi- 
ate pressures are determined from the heat diagram, and the various stages are 
then designed as separate single-stage impulse turbines, all having the same rota- 




Fig. 254. 



rtis Turbine. (General Electric Company.) 



DESIGN OF MULTI-STAGE TURBINE 



331 



tive speed. The entrance angles of the fixed intermediate blades in the Curtis 
turbine are equal to those of the absolute exit velocities of the steam. Their exit 
angles may be adjusted as desired; they may be equal to the entrance angles if 
the latter are not too acute. The greater the number of pressure stages, the 
lower is the economical limit of circumferential speed; and if the number of 
revolutions is fixed, the smaller will be the wheel. Figure 254 shows a recent 
form of Curtis turbine, with five pressure stages, each containing two rows of 
moving buckets. The electric generator is at the top. 



532. Problem. Preliminary Calculations for a Multi-stage Impulse Turbine. 
To design a 1000 (brake) hp. impulse turbine with three pressure stages, having 
two moving wheels in each pressure stage. Initial pressure, 150 lb. absolute ; 
temperature, 600° F. ; final pressure, 2 lb. absolute; entering stream angles, 20°; 
peripheral velocity, 500 ft. per second ; 1200 revolutions per minute. 

By reproducing as in Fig. 240 a portion of the Mollier heat chart, we obtain 
the expansive path A B, and the heat drop is 1316.6 — 987.5 = 329.1 B. t. u. Divid- 
ing this into three equal parts, the heat drop per stage becomes 329.1 -4- 3 = 109.7 
B. t. u. This is without correction for friction, and we may expect a somewhat 
unequal division to appear as friction is considered. To include friction in deter- 
mining the change of condition during flow through the nozzle, we lay off, in Fig. 



240, AH =109.7, HG 



AH 
10 



, and project GE, findings = 50, t = 380°, at the out- 



lets of the first set of nozzles. The velocity attained (with 10 per cent loss of 
available heat by friction) is v = 212.42 V109.7 = 2225 ft. per second. 




*^e 1 ~~u 'n f 

Fig. 255. Art. 532. — Multi-stage Velocity Diagram. 



We now lay off the velocity diagram, Fig. 249, making a = 20°, u — 500, 
v = 2225. The exit velocity x may be variously drawn; we will assume it so that 



332 APPLIED THERMODYNAMICS 

the relative angles e and /are equal, and, allowing 10 per cent for bucket friction, 
will make x = 0.9 V. For the second wheel, the angle a' is again 20°, while v f , on 
account of friction along the stationary or guide blades, is 0.9 Y. After locating 
V, if the angles e' and/' were made equal, there would in some cases be a back- 
ward impulse upon the wheel, tending to stop it, at the emergence of the jet along 
Y. On the other hand, if the angle/' were made too acute, the stream would be 
unable to get away from the moving buckets. With the particular angles and 
velocities chosen, some backward impulse is inevitable. We will limit it by mak- 
ing/' = 30°. The rotative components of the absolute velocities may be computed 
as follows, the values being checked as noted from the complete graphical solution 
of Fig. 255 : 

db = v cos 20° = 2225 x 0.93969 = 2090.81. (2080) 

cd = cz- dz = 0.9 Vcosf- u = 0.9 Fcos e - u = 0.9(2090.81 - 500) - 500 = 931.73. 

(925) 
ef = eg cos 20° = 0.9 eg * cos 20° = 0.9 x 1158 x 0.93969 = 979. (975) 

hi = km -lm = 500 - x' cos 30° = 500 - 0.9 V cos 30° 

= 500 -(0.9 x 596.2 f x 0.86603)= 36. 

ah + cd + ef- kl\ 3966 "x 500 



The work per pound of steam is then ( — ^~^ ) u 



61500 

foot-pounds, in the first stage. This is equivalent to 61,500 -=- 778 = 79.2 B. t. u. 
The heat drop assumed for this stage was 109.7 B. t. u. The heat not converted 
into work exists as residual velocity or has been expended in overcoming nozzle 
and bucket friction and thus indirectly in superheating the steam. It amounts 
to 109.7 - 79.2 = 30.5 B. t. u. 

Returning to the construction of Fig. 240, we lay off in Fig. 256 an — 79.2 
B. t. u. and project no to ko, finding the condition of the steam after passing the 
first stage buckets. Bucket friction has moved the state point from m to o, at 
which latter point Q — 1237.2,/) = 50, t = 414°. This is the condition of the steam 
which is to enter the second set of nozzles. These nozzles are to expand the steam 
down to that pressure at which the ideal (adiabatic) heat drop from the initial 
condition is 2 x 109.7 = 219.4 B. t. u. Lay off ae = 219.4, and find the line eg of 
12 lb. absolute pressure. Drawing the adiabatic op to intersect eg, we find the 
heat drop for the second stage, without friction, to be 1237.2 — 1120 = 117.2 B. t. u., 
giving a velocity of 212.42 Vll7^2 = 2299.66 ft. per second. 

* To find eg, we have 

cb = Fcos e = 2090.81 - 500 = 1590.81, bj = v sin a = 2225 x 0.34202 = 760.99, 

V=^cb 2 +bj z = ^1590M 2 + 760.99 2 = 1765, x = 0.9 V= 0.9 x 1765 = 1588.5, 

ch =zxsinf= 1588.5 sin e = 1588.5^ = 1588.5 ^^ = 685, 
J _V 1765 

eg = ^'ch 2 + hg*= ^685 2 + 931. 73 2 = 1158. 

t To find F', we have 
gf = v'sin 20° = 0.9 Tsin20° = 0.9 x 1158 x 0.34202 = 355, 
nf=ef-u = 979 - 500 = 479, V = ^nf' 2 + gf' 1 = ^479 2 + 355 2 = 596.2. 



STEAM PATH, MULTI-STAGE TURBINE 



333 



p = 150 



The complete velocity diagram must now be drawn for the second stage, fol- 
lowing the method of Fig. 255. This gives for the rotative components, ab = 2160.97, 
cd = 994.87, ef= 1032.59, Tel — 8.06. (There is no backward impulse from kl in 
this case.) The work per pound of steam is 

500(2160.97 + 994.87 + 1032.59 + 8.06) = ^^ foo ^ poundS) 

or 83.76 B. t. u. Of the available heat drop, 117.2 B. t. u., 33.44 have been ex- 
pended in friction, etc. Laying off, in Fig. 256, pq = 33.44, and projecting qr to 
meet pr, we have r as the state point for steam 
entering the third set of nozzles. Here p = 12, 
t x = 223°, Q\ = 1153.44. In expanding to the 
final condenser pressure, the ideal path is rs, 
terminating at 2 lb. absolute, and giving an un- 
corrected heat drop of Q r - Q s = 1153.44 - 1039 
= 114.44 B. t. u. The velocity attained is 
212.42 VlUAi = 2271.83 ft. per second. A third 
velocity diagram shows the work per pound of 
steam for this stage to be 63,823 foot-pounds, or 
82.04 B.t.u. We are not at present concerned 
with determining the condition of the steam at 
its exit from the third stage. 

The whole work obtained from a pound of 
steam passing through the three stages is then 
79.2 + 83.76 + 82.04 = 245.0 B. t. u. The horse 
power required is 1000 at the brake or say 
1000 - 0.8 = 1250 hp. at the buckets. This is 
1980000 



equivalent to 1250 



778 



3,181,250 B. t. 




per hour. The pounds of steam necessary per 
hour are 3,181,250 - 245.0 = 12,974. This is 
equivalent to 10.38 lb. per brake hp.-hr., a result 
sufficiently well confirmed by the test results 
given in Chapter XV. 

Proceeding now to the nozzle design, w r e 

adopt the formula F = from Art. 521. It 

V 

will be sufficiently accurate to compute cross- 
sectional areas at throats and outlets only. The 
path of the steam, in Fig. 256, is as follows : through the first set of nozzles, along 
am ; through the corresponding buckets, along mo ; thence alternately through 
nozzles and buckets along ou, ur, rv, vt. The points u, v, etc., are found as in Fig. 
240. It is not necessary to plot accurately the whole of the paths am, ou, rv; but 
the condition of the steam must be determined, for each nozzle, at that point at 
which the pressure is 0.57 the initial pressure (Art. 522). The three initial pres- 
sures are 150, 50, and 12 ; the corresponding throat pressures are 85.5, 28.5, and 
6.84. Drawing these lines of pressure, we lay off, for example, wx = ^ aw, project 



Fig. 256. Art. 532. — Steam Path, 
Multi-stage Turbine. 



334 



APPLIED THERMODYNAMICS 



xy to icy, and thus determine the state y at the throats of the first set of nozzles. The 
corresponding states are similarly determined for the other nozzles. We thus find, 

at y, p = 85.5, t = 474°, at m, p = 50, t = 380°, 
q = 1260.5 ; q = 1217.87 ; 

&tA,p = 28.5, t = 313°, at u, p = 12, x = 0.989, 
q= 1192; 5 = 1131.72; 

at B, p = 6.81, x = 0.9835, at v, p = 2, x = 0.932, 
5 = 1118; q= 1050.44. 

We now tabulate the corresponding velocities and specific volumes, as below. 
The former are obtained by taking V 



the Tumlirz formula, W 



0.5963 - - 0.256. 
P 



223.84 v q 1 — q 2 ; the latter are computed from 
Thus, at the throat of the first nozzle, 



V= 223.84 V1316.6 - 1260.5 = 1683; while W 



0.5963 46Q + 474 - 0.256 = 6.26. 

85.5 



In the wet region, the Tumlirz formula is used to obtain the volume of dry 
steam at the stated pressure and the tabular corresponding temperature ; this is 
applied to the wet vapor : W w = 0.017 + x(W - 0.017). The tabulation follows. 
Aty, V= 1683, W = 6.26 ; at m, V = 2225, W = 9.724 : 

ati F= 1507, W = 15.92; at u, V= 2299, W= 32.24; 

at B, V= 1330, W = 53.92 ; at v, V = 2271, W = 162.62. 

The value of G, the weight of steam flowing per second, is 12,974 -4- 3600 = 3.604 lb. 
For reasonable proportions, we will assume the number of nozzles to be 16 in the 
first stage, 42 in the second, and 180 in the third. The values of G per nozzle for 
the successive stages are then 3.604 - 16 = 0.22525, 3.604 - 42 = 0.08581 and 
3.604 -- 180 = 0.02002. We find values of F as follows : 



My, 

at m, 
at A, 



0.22525 x 6.26 

1683 

0.22525 x 9.724 

2225 
0.08581 x 15.92 



0.000839 ; at u, 
0.000989; at B, 
: 0.000903; at v, 



0.08581 x 32.24 

2299 
0.02002 x 53.92 

1330 
0.02002x162.62 



= 0.001205 ; 
= 0.000809 ; 
= 0.00144. 



1507 2271 

Completing the computation as to the last set of nozzles only, the throat 
area is 0.000809 sq. ft., that at the outlet being 0.00144 sq. ft. These corre- 
spond to diameters of 0.385 and 
0.515 in. The taper may be uniform 
from throat to outlet, the sides mak- 
ing an angle of 10°. This requires 
a length from throat to outlet of 
(0.515 - 0.385) -f- 2 tan 5° = 0.742 in. 
The length from inlet to throat may 
be one fourth this, or 0.186 in., the 
edge of the inlet being rounded. 
The nozzle is shown in Fig. 257. 
The diameter of the bucket wheels at mid-height is obtained from the rotative 
speed and peripheral velocity. If d be the diameter, 

3*1416 d x 1200 = 60 x 500, or d = 7.98 feet. 




Fig. 257. Art. 532. —Third Stage Nozzle. 



PRESSURE TURBINE 



335 



The forms of bucket are derived from the velocity diagrams. For the first 
stage, we proceed as in Art. 530, using the relative angles e and /given in Fig. 255 
for determining the angles of the backs of the moving blades, and the absolute 
angles for determining those of the stationary blades. 

533. Utilization of Pressure Energy. Besides the energy of impulse 
against the wheel, unaccompanied by changes in pressure, the steam may 
expand while traversing the backets, producing work by reaction. This 
involves incomplete expansion in the nozzle, and makes the velocities of 
the discharged jets much less than in a pure impulse turbine. Lower 
rotative speeds are therefore practicable. Loss of efficiency is avoided by 
carrying the ultimate expansion down to the condenser pressure. In the 
pure pressure turbine of Parsons, there are no expanding nozzles ; all of 
the expansion occurs in the buckets (Art. 524). (See Fig. 247.) Here 
the whole useful effort is produced by the reaction of the expanding steam 
as it emerges from the working blades to the guide blades. No velocity is 
given up daring the passage of the steam ; the velocity is, in fact, increasing, 
hence the name reaction turbine. The impulse turbine, on the contrary, 
performs work solely because of the force with which the swiftly moving 
jet strikes the vane. It is sometimes called the velocity turbine. Turbines 
are farther classified as horizontal or vertical, according to the position of 
the shaft, and as radial flow or axial flow, according to the location of the 
successive rows of buckets. Most pressure turbines are of the axial flow 
type. 



534- Design of Pressure Turbine. The number of stages is now large. The 
heat drop in any stage is so small that the entering velocity is no longer negligible. 
The velocities which determine the rate of conversion of heat into work will vary 
during the passage of steam, being reduced by friction and increased by expansion : 
the latter being provided by appropriately shaping the buckets. We may assume 

,A a reduction of heat drop by friction 
— say 25 per cent — and plot the ex- 
pansive path as in Fig. 240. This 
permits of determining the pressure, 
volume, and quality at any tempera- 
ture. 

In Fig. 259, let the turbine have 
four drums, FC, CD, DB, BO. The 
peripheral speeds of these drums may 
vary from 130 to 350 feet per second. 
We will now assume absolute veloci- 
ties for the steam entering each set of moving blades, as along EA. It is cus- 
tomary to allow these velocities to range from li to 3| times the peripheral speed 
of the drums ; they should increase quite rapidly toward the last stages of ex- 
pansion. Knowing the steam velocity and peripheral velocity for any state like 




Pressure Turbine. 



336 



APPLIED THERMODYNAMICS 



Z, we construct a velocity diagram as in Fig. 249, choosing appropriate angles of 
entrance and exit. In ordinary practice, the expansion in the buckets is sufficient, 

notwithstanding friction, to make the rela- 
tive exit and absolute entrance angles and 
velocities about equal. In such case, we 
have the simple graphical construction of 
Fig. 260. 

Since ab = be, db = be, and ad = ec, we 
obtain 

ad(hc + hd) 
32.2 




work = HM±M 

32.2 



Fig. 260. Art. 534, Prob. 18. — Velocity 
Diagram, Pressure Turbine. 

_ dg 



Drop the perpendicular bh, and with h 
as a center describe the arc aj. Draw 
dg perpendicular to ac. Then 

-2 



d 9 



work = - r ^- foot-pounds, 



32.2 



ad x dc = ad(dh + he), and 
dg N 2 



158.3 



B. t. 



In the general case, the work may be computed as in Art. 532. This result 
represents the heat converted into work at a stage located vertically in line with 
the point Z, Fig. 259. Let this heat be laid off to some convenient scale, as GH. 
Similar determinations for other states give the heat drop curve IJKHLMNOP. 
The average ordinate of this curve is the average heat drop or work done per 
stage. If we divide the total heat drop obtained by the average drop per stage, 
we have the number of stages, the nearest whole number being taken.* The 
diameter of any drum at mid-height of buckets is computed from the peripheral 
velocity and number of revolutions per minute. 

535. Details. The bucket spacing and heights must be such as to give room 
for the passage of the necessary volume of steam, which depends upon the turbine 
output and varies with the stage of expansion attained. The blade heights should 
be at least 3 per cent of the drum diameter, to avoid excessive leakage over their 
tips. The clearance over tips in inches should be from 0.01 d to 0.008 d, where d 
is the drum diameter in feet. Blade widths vary from f to 1^ in., with center to 
center spacing of from 1\ to 4 in. Blade angles are obtained from the velocity 
diagram. 

If A is the angle made between the steam leaving the guide vanes and the 
plane of the wheel, and c is the absolute velocity of the stream, the axial com- 
ponent of this velocity is c sin A. Let the number of buckets on a wheel (stage) 
be n, their height I, and their spacing e. Without allowance for thickness of 
buckets, the area for passage of steam would be nel; the usual thickness of 
buckets will reduce this to f nel. The volume of steam discharged per second will 
then be \nelc sin^l = Gw, in which G is the weight of flow per second and w the 
specific volume, which varies while the steam is traversing a single row of buckets. 

* Dividing the total heat drop at a state in a vertical line through C by the average 
drop per stage from F to C, we have the number of stages on the first drum. 



PRESSURE TURBINE 



337 



Since ne is the circumference of the wheel = ird, where d is the diameter, we have 
§ irdlc sin A = Gw. 

The successive drum diameters frequently have the ratio V2 : 1 (21). 

Specimen Case 

To determine the general characteristics of a pressure turbine operating be- 
tween pressures of 100 and 3.5 lb., with an initial superheat of 300° F., the heat 
drop being reduced 25 per cent by friction. There are to be 3 drums, and the heat 
drop is to be equally divided between the drums. The peripheral speeds of the 
successive drums are 160, 240, 320 ft. per second. The relative entrance and 
absolute exit velocities and angles are equal : the absolute entrance angle is 20°. 
The turbine makes 300 r. p. m. and develops 2500 kw. with losses between buckets 
and generator output of 65 per cent. 



* oo « o 


1.784 
1.788 
1.792 
1.796 


ENTROPY 

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Fig. 260 a. Art. 535. — Expansion Path, Pressure Turbine. 



338 



APPLIED THERMODYNAMICS 



In Fig. 260 a, the expansive path is plotted on a portion of the total heat- 
entropy diagram. The total heat drop is shown to be 1342 — 1130 = 212 B. t. u., 
and the heat drop per drum is 212 +- 3 = 70f B. t. u. In Fig. 260 b, lay off to any 
scale the equal distances ab, be, cd, and the vertical distances ae, bg, ci, rep- 
resenting the drum speeds. Lay off also ak, bm, co, equal respectively to 
11 x (ae, bg, ci), and al, bn, cp, equal respectively 
to 3 i times these drum speeds. The curve 




b c 

Fig. 260 b. Art. 535. — Elements of Pressure Turbine. 



of entrance absolute velocities is now assumed, so as to lie wholly within the area 
lisntpuvowmx. Figure 260 c shows the essential parts of the velocity diagram 
for the stages on the first drum. Here ab represents aq in Fig. 260 b, ad represents 



ae, the angle bad is 20°, and 



/ de 



/ 279.7 V 
U58.3/ 



3.12 B. t. u. is the heat drop 



V158.3. 
for the first stage in the turbine. Making ac represent by and drawing dc, eh, of, 



we find 



(JLV = 

V158.3/ 



/ 304.7 



3.70 B. t. u. as the heat drop for the last stage on 



V158.3, 
the first drum. For intermediate stages between these two, we find, 



Initial Absolute 
Velocity 


Ordinate from 
d 


Heat Drop, 

B. T. IT. 


ab = 350 


de = 279.7 


3.12 


3561 


282.8 


3.20 


3621 


285.9 


3.26 


368f 


289.0 


3.34 


375 


292.1 


3.40 


3811 


295.3 


3.48 


387| 


298.4 


3.56 


393 1 


301.5 


3.63 


ac = 400 


df= 304.7 


3.70 



PRESSURE TURBINE 



339 



In Fig. 260 b, we now divide the distance ab into 8 equal parts and lay off to 
any convenient vertical scale the heat drops just found, obtaining the heat drop 
curve zA. The average ordinate of this curve is 3.41 and the number of stages on 

21 (nearest whole number). The number of stages 



the first drum is 70f -r- 3.41 




Fig. 260 c. Art. 535. — Velocity Diagram, Pressure Turbine. 



on the other drums is found in the same way, the peripheral velocity ad, Fig. 
260 c, being different for the different drums. The diameter d of the first drum is 
given by the expression 

60 x 160 



300 ird = 60 x 160 or d = 

3.1416 x 300 

The weight of steam flowing per second is 
2500 x 1.34 x 2545 



10.2 ft. 



0.65 x 212 x 3600 



= 17.1 lb. 



340 APPLIED THERMODYNAMICS 

In the first stage of the first drum, the condition of the steam at entrance to 
the guide blades is (Fig. 260 a) H = 1342, p = 100; at exit from the moving 
blades, it is H = 1338.59, p = 98. From the total heat-pressure diagram, or by 
computation, the corresponding specific volumes are 6.5 and 6.6. The volumes of 
steam flowing are then 6.5 x 17.1 = 111 and 6.6 x 17.1 = 113 cu. ft. per second. 
The absolute steam velocities are (Fig. 260 b) 350 and 356^ ft. per second. The 
axial components of these velocities (entrance angle 20°) are 0.34202 x 350 = 120, 
and 0.34202 x 356| = 122. The drum periphery is 10.2 x 3.1416 = 32 ft. If the 
blade thicknesses occupy i this periphery and the width for steam passage between 
the buckets is constant, the width for passage of steam is f x 32 = 21.33 ft. and 

the necessary height of fixed buckets is = 0.434 ft. or 5.2 in. at the 

J & 21.33 x 120 

beginning of the stage and — — — — - = 0.434 ft. or 5.2 in. at the end. The 

fixed blade angles are determined by the velocities be and ab, Fig. 260: those of 
the moving blades by bd and be. There is no serious error involved in taking the 
velocity and specific volume as constant throughout a stage. The height of the 
moving buckets should of course not be less than that of the guide blades ; this 
may be accomplished by increasing the thickness of the former. 

It should be noted that the velocities indicated by the curve qr, Fig. 260 b, are 
those of the steam at exit from the fixed blades and entrance to the moving blades. 
The diagram of Fig. 260 gives the absolute velocity of the steam entering the next 
set of fixed blades. 



Commercial Forms of Turbine. 

536. De Laval; Stumpf. Figure 235 illustrates the principle of the De Laval 
machine, the working parts of which are shown in Fig. 261. Entering through 
divergent nozzles, the steam strikes the buckets around the periphery of the wheel 
b. The shaft c transmits power through the helical pinions a, a, which drive the 
gears e, e, e, e, on the working shafts/,/. The wheel is housed with the iron cas- 
ing g. This is a horizontal single-stage impulse turbine, with a single wheel. 
Its rotative speed is consequently high ; in small units, it reaches 30,000 r. p. m. 
It is built principally in small sizes, from 5 to 300 h. p. The nozzles make angles 
of 20° with the plane of the wheel ; the buckets are symmetrical, and their angles 
range from 32° to 36°, increasing with the size of the unit. For these proportions, 
the most efficient values of u would be about 950 and 2100 for absolute steam veloci- 
ties of 2000 and 4400 feet per second, respectively; in practice, these speeds are 
not attained, u ranging from 500 to 1400 feet per second, according to the size. 
The high rotative speeds require the use of gearing for most application?. The 
helical gears used are quiet, and being cut right- and left-hand respectively they 
practically eliminate end thrust on the shaft. The speed is usually reduced in the 
proportion of 1 to 10. The high rotative speeds also prevent satisfactory balanc- 
ing, and the shaft is, therefore, made flexible ; for a 5-hp. turbine, it is only f 
inch in diameter. The bearings h, j are also arranged so as to permit of some 
movement. The pressure of steam in the wheel case is that of the atmosphere or 
condenser, all expansion occurring in the nozzle. A centrifugal governor controls 



DE LAVAL TURBINE 



341 




342 



APPLIED THERMODYNAMICS 



the speed by throttling the steam supply and by opening communication between 
the wheel case and atmosphere when necessary. 

The nozzles of the De Laval turbine are located as in Fig. 235. Those of the 
Stumpf, another turbine of this class, are tangential, while the buckets are of the 
Palton form (Fig. 252), and are milled in the periphery of the wheel. A very 
large wheel is employed, the rotative speeds being thus reduced. In a late form 
of the Stumpf machine, a second stage is added. The reversals of direction are so 
extreme that the fluid friction must be excessive. 

537. Curtis Turbine. This is a multi-stage impulse turbine, the principle of 
operation having been shown in Fig. 245. In most cases, it is vertical ; for marine 

applications, it is necessarily made 
horizontal. Figure 262 illustrates 
the stationary and moving blades 
and nozzles. Steam enters through 
the nozzle A , strikes a row of mov- 
ing vanes at a, passes from them 
through stationary vanes B to 
another row of moving vanes at e, 
then passes through a second set 
of expanding nozzles at h to the 
next pressure stage. This particu- 
lar machine has four pressure 
stages with two sets of moving 
buckets in each stage. The direc- 
tion of flow is axial. The number 
of pressure stages may range from 
two to seven. From two to four 
velocity stages (rows of moving 
buckets) are used in each pressure 
stage. In the two-stage machine, 
the second stage is disconnected 
when the turbine runs non-con- 
densing, the exhaust from the first 
stage being discharged to the at- 
mosphere. Governing is effected 
by automatically varying the number of nozzles in use for admitting steam to the 
first stage. A step bearing carries the whole weight of the machine, and must be 
supplied with lubricant under heavy pressure ; an hydraulic accumulator system is 
commonly employed. 

538. Rateau Turbine. This is a horizontal, axial flow, multi-stage impulse 
turbine. The number of pressure stages is very large — from twenty-five upward. 
There is one velocity stage in each pressure stage. Very low speeds are, therefore, 
possible. Figure 263 shows the general arrangement; the tranverse partitions e, e 
form cells, in which revolve the wheels/, /; the nozzles are merely slots in the 
partitions. The blades are pressed out of sheet steel and riveted to the wheel. 
The wheels themselves are of thin pressed steel. 




Fig. 262. Art. 537. — Curtis Turbine. 



APPLICATIONS OF TURBINES 



343 




Fig. 263. Art. 538. -Rateau Turbine. 

539. Westinghouse-Parsons Turbine. This is of the axial flow pressure type, 
and horizontal. The steam expands through a large number of successive fixed 
and moving blades. In Fig. 264, the steam enters at A and passes along the vari- 
ous blades toward the left; the movable buckets are mounted on the three drums, 
and the fixed buckets project inward from the casings. The diameters of the 
drums increase by steps ; the increasing volume of the steam within any section is 
accommodated by varying the bucket heights. The balance pistons P, P, P are 
used to counteract end thrust. The speed is fairly high, and special provision 
must be made for it in the design of the bearings. Governing is effected by inter- 
mittently opening the valve V\ this valve is wide open whenever open at all. 

The length of this machine is sometimes too great for convenience. To over- 
come this, the " double-flow " turbine receives steam near its center, through 
expanding nozzles which supply a simple Pelton impulse wheel. This utilizes 
a large proportion of the energy, and the steam then flows in both directions 
axially, through a series of fixed and moving expanding buckets. Besides reduc- 
ing the length, this arrangement practically eliminates end thrust and the neces- 
sity for balance pistons. 



540. Applications of Turbines. Turbo-locomotives have been experimented 
with in Germany ; the direct connection of the steam turbine to high-pressure 
rotary air compressors has been accomplished. In stationary work, the direct 
driving of generators by turbines is common, and the high rotative speeds of the 
latter have cheapened the former. At high speeds, difficulties may be experi- 
enced with commutation; so that the turbine is most successful with alternating- 
current machines. When driving pumps, turbines permit of exceptionally high 
lifts with good efficiencies for the centrifugal type, and low first costs. For low- 
pressure, high-speed blowers, the turbine is an ideal motor. The outlook for a gas 
turbine is not promising, any gas cycle involving combustion at constant pressure 
being both practically and thermodynamically inefficient. 

The objections to the turbine in marine application have arisen from the high 



344 



APPLIED THERMODYNAMICS 




EXHAUST STEAM TURBINES 345 

speed and the difficulty of reversing. A separate reversing wheel may be em- 
ployed, and graduation of speed is generally attained by installing turbines in 
pairs. A small reciprocating engine is sometimes employed for maneuvering at 
or near docks. Since turbines are not well adapted to low rotative speeds, they 
are not recommended for vessels rated under 15 or 16. knots. The advantages of 
turbo-operation, in decreased vibration, greater simplicity, smaller and more deeply 
immersed propellers, lower center of gravity of engine-room machinery, decreased 
size, lower first cost, and greater unit capacity without excessive size, have led to 
extended marine application. The most conspicuous examples are in the Cunard 
liners Lusitania and Mauretania. The former has two high-pressure and two low- 
pressure main turbines, and two astern turbines, all of the Parsons type (22). 
The drum diameters are respectively 96, 140, and 104 in. An output of 70,000 hp. 
is attained at full speed. 

541. The Exhaust-steam Turbine. From the heat chart, Fig. 177, it is 
obvious that steam expanding adiabatically from 150 lb. absolute pressure and 
600° F. to 1.0 lb. absolute pressure transforms into work 365 B. t. u. It has been 
shown that in the ordinary reciprocating engine such complete expansion is unde- 
sirable, on account of condensation losses. The final pressure is rarely below 7 lb. 
absolute, at which the heat converted into work in the above illustration is only 
252 B. t. u. The turbine is particularly fitted to utilize the remaining 113 B. t. u. 
of available heat. The use of low-pressure turbines to receive the exhaust steam 
from reciprocating engines, has, therefore, been suggested. Some progress has 
been made in applying this principle in plants where the engine load is intermit- 
tent and condensation of the exhaust would scarcely pay. With steel mill en- 
gines, steam hammers, and similar equipment, the introduction of a low-pressure 
turbine is decidedly profitable. The variations in supply of steam to the turbine 
are offset by the use of a regenerator or accumulator, a cast-iron, water-sprayed 
chamber having a large storage capacity, constituting a "fly wheel for heat," and 
by admitting live steam to the turbine through a reducing valve. When a sur- 
plus of steam reaches the accumulator, the pressure rises ; as soon as this falls, 
some of the water is evaporated. The maximum pressure is kept low to avoid 
back pressure at the engines. A steam consumption by the turbine as low as 
35 lb. per brake hp.-hr. has been claimed, with 15 lb. initial absolute pressure and 
a final vacuum of 26 in. Other good results have been shown in various trials 
(23). Wait (24) has described a plant at South Chicago, 111., in which a 42 by 
60 double cylinder, reversible rolling-mill engine exhausts to an accumulator at a 
pressure 2 or 3 lb. above that of the atmosphere. This delivers steam at about 
atmospheric pressure to a 500 kw. Kateau turbine operated with a 28-in. vacuum. 
The steam consumption of the turbine was about 35 lb. per electrical hp.-hr., 
delivered at the switchboard. 

The S.S. Turbinia, in 1897, was fitted with low-pressure turbines receiving the 
exhaust from reciprocating engines and operating between 9 lb. and 1 lb. absolute. 
One third of the total power of the vessel was developed by the turbines, although 
the initial pressure was 160 lb. 

542. Commercial Considerations. The best turbines, in spite of their thermo- 
dynamically superior cycle, have not yet equalled in efficiency the best reciprocat- 



346 APPLIED THERMODYNAMICS 

ing engines, both operating at full load. The average turbine is more economical 
than the average engine ; and since the mechanical and fluid friction losses are 
disproportionately large, it seems reasonable to expect improved efficiencies as 
experimental knowledge accumulates. 

The turbine is cheaper than the engine ; it weighs less, has no fly wheel, 
requires less space and very much' less foundation. It can be built in larger units 
than a reciprocating cylinder. Power house buildings are cheapened by its use ; the 
cost of attendance and of sundry operating supplies is reduced. It probably depre- 
ciates less rapidly than the engine. The wide range of expansion makes a high 
vacuum desirable ; this leads to excessive cost of condensing apparatus. Similarly, 
superheat is so thoroughly beneficial in reducing steam friction losses that a con- 
siderable investment in superheaters is necessary. The turbine must have a direct 
connected balanced load ; so that the cost of generators must often be included in 
the initial expense, although otherwise unnecessary. The choice as between the 
turbine and the engine must be determined with reference to all of the condi- 
tions, technical and commercial, including that of load factor. Turbine economy 
cannot be measured by the indicator, but must be determined at the brake or 
switchboard and should be expressed on the heat unit basis (B. t. u. consumed per 
unit of output per minute). 

(1) Trans. Inst. Engrs. and Shipbuilders in Scotland, XLVI, V. (2) Berry, 
The Temperature-Entropy Diagram, 1905. (3) To show this, put the expression in 

y 

the brace equal to m, and make — = ; then — = ( y + ) , which may be solved 

dp p \ 2 J 

for any given value of y. (4) Thesis, Polytechnic Institute of Brooklyn, 1905. 

(5) Thomas, Steam Turbines, 1906, 89. (6) Proc. Inst. Civ. Eng., CXL, 199. 

(7) Zeits. Ver. Deutsch. Ing., Jan. 16, 1904. (8) Rankine, The Steam Engine, 1897, 

344. (9) Experimental Besearches on the Flow of Steam, Brydon tr. ; Thomas, op. cit., 

106. (10) Thomas, op. cit., 123. (11) Engineering, XIII (1872). (12) Trans. 

A. S. M. E., XI, 187. (13) Mitteil. uber Forschungsarb., XVIII, 47. (14) Practice 

and Theory of the Injector, 1894. (15) Peabody, Thermodynamics, 1907, 443. 

(16) Trans. A. S. 31. E., XXVII, 081. (17) Stodola, Steam Turbines. (18) The 

Steam Engine, 1905, I, 170. (19) Technical Thermodynamics, Klein tr., 1907: I, 

225: II, 153. (20) Trans. A. S. M. E., XXVII, 081. (21) See H. F. Schmidt, in 

The Engineer (Chicago), Dec. 16, 1907: Trans. Inst. Engrs. and Shipbuilders in 

Scotland, XLXIX. (22) Power, November, 1907, 770. (23) Trans. A. S. M. E., 

XXV, 817 : Ibid, XXXII, 3, 315. (24) Proc. A. I. E. E., 1907. 

OUTLINE OF CHAPTER XIV 

The turbine utilizes the velocity energy of a jet or stream of steam. 

Expansion in a nozzle is adiabatic, but not isentropic ; the losses in a turbine are due 

to residual velocity, friction of steam through nozzles and buckets and mechanical 

friction. 

E + PW+ — = e + pw + — , or — = q— Q, approximately ; 
2<7 2g 2g 



whence V= 223.84 Vq - Q. 

The complete expansion secured in the turbine warrants the use of exceptionally high 
vacuum. 



THE STEAM TURBINE 347 

Nozzle ' friction decreases the heat converted into work and the velocity attained ; 

V- 212.42 Vq^ij. 

The heat expended in overcoming friction reappears in drying or superheating the 
steam. 

W P 

F= G — , which reaches a minimum at a definite value of — • For steam, this value 
V p 

is about 0.57. If the discharge pressure is less than 0.57 p, the nozzle converges to 

a "throat" and afterward diverges. 
The multi-stage impulse turbine uses lower rotative speeds than the single stage. 
The diverging sides of the nozzle form an angle of 10° ; the converging-portion may be 

one fourth as long. 
Steam consumption per Ihp.-hr. — 2545 -h E(q — Q). 
The rotative components of the absolute velocities determine the work ; the -relative 

velocities determine the (moving) bucket angles. Bucket friction may decrease 

relative velocities by 10 per cent during passage. Work = (y cos a ■£ Y cos p) - . 

9 . 

Efficiency = E = Work -h 778 (q — Q). Bucket angles may be adjusted to equalize 

end thrust, to secure maximum work, or may be made equal. 
For a right-angled stream change, maximum efficiency is 0.50 ; with complete reversal, 

it is 1.00. With practicable buckets, it is always less than 1.0. • 

The backs of moving buckets are made tangent to the relative stream velocities. 
The angles of fixed blades are determined by the absolute velocities. 
In the pure pressure turbine, expansion occurs in the buckets. No nozzles are used. - 
Turbines may be horizontal or vertical, radial or axial flow, impulse or pressure type., 

In designing a pressure turbine, -= 0.30 to 0.75. The heat drop at any stage may 

v ■ ;■* ■ ■ • - y 

equal [ — ^— ) , Fig. 260. The number of stages is the quotient of the whole heat 



158.3 

drop, corrected for friction, by the mean value of this quantity. Friction through 
buckets may be from 20 to 30 per cent. The accumulated heat drop to any stage 
is ascertained and the condition of the steam found as in Fig. 240. 
Commercial forms include the De Laval, single-stage impulse : 

Stumpf, single- or two-stage impulse, with Pelton buckets. 
Curtis, multi-stage impulse, usually vertical, axial flow. 
Bateau, multi-stage impulse, axial flow, horizontal, many stages. 
Westinghouse-Parsons, pressure type, axial flow, horizontal ; sometimes of the 
" double flow " form. 
Marine applications involve some difficulty, but have been satisfactory at high speeds. 
The turbine may utilize economically the heat rejected by a reciprocating engine. A 

regenerator is sometimes employed. 
The best recorded thermal economy has been attained by the reciprocating engine ; 
but commercially the turbine has many points of superiority. 

PROBLEMS 

1 . Show on the TN diagram the'ideal cycle for a turbine operating between pressure 
limits of 140 lb. and 2 lb., with an initial temperature of 500° F. and adiabatic 
(isentropic) expansion. What is the efficiency of this cycle ? 



348 APPLIED THERMODYNAMICS 

2. In Problem 1, what is the loss of heat contents and the velocity ideally attained ? 

3. In Problem 1, how will the efficiency and velocity be affected if the initial 
pressure is 150 lb ? If the initial temperature is 600° F. ? If the final pressure is 1 lb. ? 

4. Solve Problems 1, 2, and 3, making allowance for friction as in Art. 519. 

5. Compute analytically, in Problem 3, first case, the condition of the steam after 
expansion, as in Art. 520, assuming the heat drop to have been decreased 10 per cent 
by friction. 

6. An ideal reciprocating engine receives steam at 150 lb. pressure and 550° F., 
and expands it adiabatically to 7 lb. pressure. By what percentage would the efficiency 
be increased if the steam were afterward expanded adiabatically in a turbine to 1.5 lb. 
pressure ? 

7. Steam at 100 lb. pressure, 92 per cent dry, expands to 16 lb. pressure. The loss 
of heat drop due to friction is 10 per cent. Compute the final condition and the velocity 
attained. 

8. In Problem 5, find the throat and outlet diameters of a nozzle to discharge 
1000 lb. of steam per hour, and sketch the nozzle. 

P 

9. Check the value — = 0.5274 for maximum flow in Art. 522. 

P 

10. Check the equation of flow of a permanent gas, in Art. 522. 

11. If the efficiency in Problem 5, from steam to shaft, is 0.60, find the steam con- 
sumption per brake hp.-hr , and the thermal efficiency. 

12. In Problem 5, let the peripheral speed be u = 480, the angle a = 20°, and find 
the work done per pound of steam in a single-stage impulse turbine (a) with end thrust 
eliminated, (b) with equal relative angles. Allow a 10 per cent reduction of relative 
velocity for bucket friction. 

13. In Problem 12, Case (&), what is the efficiency from steam to work at the 
buckets? (Item E, Art. 526.) 

14. Sketch the bucket in Problem 12, Case (6), as in Art. 530. 

15. Compute the wheel diameters and design the first-stage nozzles and buckets for 
a two-stage impulse turbine, with two moving wheels in each stage, as in Art. 532, 
operating under the conditions of Problem 5, the capacity to be 1500 kw., the entering 
stream angles 20°, the peripheral speed 600 ft. per second, the speed 1500 r. p. m., the 
heat drop reduced 0.10 by nozzle friction. Arrange the bucket angles to give the highest 
practicable efficiency,* the stream velocities to be reduced 10 percent by bucket friction. 
State the heat unit-consumption per kw. -minute. 

16. In Problem 5, plot by stages of about 10 B. t. u. the NT expansion path in a 
pressure turbine in which the heat drop is decreased 0.25 by bucket friction. 

17. In Problem 16, the drums have peripheral speeds of 150, 250, 350. Construct a 
reasonable curve of steam velocities, as in Fig. 259, the velocity of the steam entering 
the first stage being 400 ft. per second, and the corrected heat drop through, the drums 
being equally divided. 

18. In Problem 17, let the absolute entrance angles be 20°, and let the velocity 
diagram be as in Fig. 260. Find the work done in each of six stages along each drum. 
Find the average heat drop per stage, and the number of stages in each drum, the total 
heat drop per drum having been obtained from Problem 16. 

*The angle /must not be less than 24° in any case. 



THE STEAM TURBINE 349 

19. The speed of the turbine in Problem 18 is 400 r. p. m. Find the diameter of each 
drum. 

20. In Problems 16-19, the blades are spaced 2" centers. The turbine develops 
1500 kw. Find the heights of the moving blades for one expansive state, assuming 
losses between buckets and generator of 45 per cent. Design the moving bucket. 

21. Sketch the arrangement of a turbine in which the steam first strikes a Pelton 
impulse wheel and then divides ; one portion traveling through a three-drum pressure 
rotor axially, the other through a two-pressure stage velocity rotor with three rows of 
moving buckets in each pressure stage, also axially, the shaft of the velocity turbine 
being vertical. 

22. Compare as to effect on thermal efficiency the methods of governing the 
De Laval, Curtis, and Westinghouse-Parsons turbines. 

23. Determine whether the result given in Art. 541, reported for the S.S. Turbinia, 
is credible. 



CHAPTER XV 

RESULTS OF TRIALS OF STEAM ENGINES AND STEAM TURBINES 

543. Sources. The most reliable original sources of information as to con- 
temporaneous steam economy are the Transactions or Proceedings of the various 
national mechanical engineering societies (1). The reports of the Committee of 
the Institution of Mechanical Engineers on Marine Engine Trials are of special 
interest (2). The Alsatian experiments on superheating have already been re- 
ferred to (Art. 443). The works of Barrus (3) and of Thomas (4) present a mass 
of results obtained on reciprocating engines and turbines respectively. The 
investigations of Isherwood are still studied (5). 

544. Limiting Efficiencies. Neither the engine nor the turbine can, in prac- 
tice, give an efficiency equal to that of the corresponding Clausius cycle. Actual 
tests show efficiency ratios ranging usually between 50 and 75 per cent, but occa- 
sionally overlapping one of these limits. The Clausius efficiency depends solely 
upon the temperature limits, so that we may expect engine efficiencies to be 
improved by high pressures or superheats and good vacua. The actual engine is 
subject to various additional modifying conditions ; in general, for a given tem- 
perature range in the cylinder, we may find the efficiency to be improved by well- 
designed valves, fairly low terminal pressures and reasonably wide ratios of 
expansion, jackets (unless the steam is superheated), and multiple expansion. 
Since engines are generally governed by varying the ratio of expansion, we may 
find that steady loads, as in pumping service, which lead to uniform ratios of 
expansion, are also associated with maximum efficiencies. 

545. Basis for Rating. The heat unit basis is the only proper standard for 
comparing the performance of engines operating under dissimilar conditions. On 
account of the uncertainty which has existed as to the specific heat of superheated 
steam, various constant or variable values have been employed in computing the 
results of trials in which superheat was used. These lead to results not strictly 
comparable, although the error can seldom be of much consequence. For the 
present, at least, trials made with superheated steam should be so reported that 
the correction for superheat may be independently made by any one, using such 
values as he prefers for the specific heat. 

546. Non-condensing Trials. Usual steam rates (pounds of dry steam per 
Ihp.-hr.) range from 21.5 (with jackets) up to 38, in good simple engines, when 
new. A fair rate with an unjacketed cylinder is 30 ; poor engines, such as direct- 

350 



RESULTS OF TRIALS 



351 



acting steam pumps, without expansion, show steam rates up to 319 pounds (6) 
or more. Dean (7) quotes a number of tests on high-speed single-valve and four- 
valve engines, showing that the best efficiency is obtained at rather low ratios of 
expansion, and that the economy falls off rapidly with wear. Mechanical efficiencies 
range from 75 to 90 per cent; the combined mechanical efficiency of a small direct- 
connected engine and generator of these types may be taken under ordinary con- 
ditions at 75 per cent. Steam pressures seldom range above 100 lb.* 

Multiple-expansion engines seldom run non-condensing. 
Willans found for compounds steam rates of 19.14 to 23; for 
triples a minimum rate of 18.5 has been obtained (8). 

547. Simple Condensing Engines. As early as 1840, the 
famous Cornish pumping engines, with expansion ratios from 
1.5 to 3.5, gave, under the best conditions, steam rates of 16.5 
to 24 lb, (9). No improvement has been made over these 
figures ; usual rates range from 16.9 (with jackets) to 24.2. 
The famous Leavitt pumping engine at Lawrence gave 16.5, 
with 120 lb. initial pressure, 16 expansions, and 12 r. p. m. 



548. Compound Condensing Engines. Steam rates range 
downward from 21 lb., in good types well operated. In 1878, 
the Corliss Pawtucket pumping engine gave 13.7 lb. with 120 lb. 
steam pressure. Pressures now range up to 175 lb. Rock- 
wood's high ratio compound (Art. 480), with 150 lb. steam 
pressure, gave a rate of 12.45. Jacobus (10) tested a Rice and 
Sargent engine which gave 12.10 lb. This ran, at 1501b. steam 
pressure and 28 inches of vacuum, at 120 r. p. m. The com- 
bined diagrams showing the effect of the jackets and reheaters 
appear in Fig. 265. A curve showing the steam consumption 
at various loads is given in Fig. 266. A heat unit consumption 
of 222 B. t. u. per Ihp.-minute was reached at normal load of 
700 hp. : the economy held up well at heavy overloads, a point 

of much com- 
mercial impor- 
tance. A 250 hp. 
Van den Ker- 
chove engine 
(11) at 126 
r. p. m., 130 lb. 
- pressure, and 
32 expansions 
gave a rate of 




Fig. 265. Art. 548. — Rice and Sargent Engine Diagrams. 



11.98 lb. A Westinghouse engine of 5400 hp. at 185 lb. pressure gave 11.93 lb. 
Barrus and Rockwood, with 175 lb. pressure, obtained the best rate thus far re- 
ported — 11.22 lb. — on another " wide ratio " compound. 



* Pressures given in .this chapter, unless otherwise specified, are gauge pressures. 



352 



APPLIED THERMODYNAMICS 



549. Triple-Expansion Condensing Engines. The experimental engine at 
the Massachusetts Institute of Technology gave the following heat unit consump- 
tions per Ihp. per minute: 319 as a compound without jackets: 274 as a triple 
without jackets ; 261 with jackets on heads ; 239 with jackets on whole of cylinders 



15 


V 






























\ 


N 
































V o 
























19 




















t 








o 




< 


\ 




o 
© 
to 






© 


1 



INDICATED HORSE POWER 



Fig. 266. Art. 548. — Test of Rice and Sargent Engine. 

and receiver ; 233 with jackets on cylinders only. Steam rates, in practice, range 
downward from 16 lb., with steam pressures usually under 200 lb. Willans ob- 
tained 12.74 lb. ; Schroter, 12.2 and 12.65 lb. A very small engine has given 
12.68 lb. (12). A rate of 12.5 lb. would be extremely good in ordinary mill 
service. With only 124 lb. pressure, and a merely fair vacuum, Cooley (13) ob- 
tained 12.65 lb. on a 15,000,000-gallon Nordberg pumping engine. At an even 
lower pressure (121.4 lb.), an Allis pumping engine gave an 11.68 lb. rate. 
Laird reports (14) for two 10,000,000-gallon Allis pumps at 136.5 lb. pressure, an 
average rate of 11.63 lb. or 216.7 B. t. u. per Ihp. per .minute, with a mechanical 
efficiency of 94.6 per cent. A 20,000,000-gallon Snow pump tested by Goss (15) 
with 155 lb. pressure, gave a rate of 11.38 lb. and 94 per cent mechanical effi- 
ciency. The Leavitt engine gave 11.22 lb. with 176 lb. steam pressure. The best 
rate recorded at its date on saturated steam was made by the Allis pumping 
engine at Hackensack, N.J., about 1904. This used a pressure of 188 lb., 33 
expansions, and ran at 30 r. p. m. ; its steam rate was 11.05 lb., or 211 B. t. u. 
per Ihp. per minute. The best triple has thus only slightly excelled the best 
compounds. 



550. Quadruple Engines. 

\. 



The 




Fig. 267. 



Art. 550. — Nordberg Engine 
Diagrams. 



most economical performances on record 
with saturated steam have been made 
in quadruple-expansion engines. The 
Nordberg pumping engine at Wildwood 
(16) although of only 6,000,000 gal. 
capacity (712 horse power), and jack- 
eted on barrels of cylinders only, gave 
a heat consumption of 185.96 B. t. u. 
with 200 lb. initial pressure and only a 
fair vacuum. The high efficiency was 
obtained by drawing off live steam 
from each of the receivers and trans- 
ferring its high-temperature heat direct 
to the boiler feed water by means of 
coil heaters. Heat was thus absorbed 
more nearly at the high temperature 



TESTS WITH SUPERHEATED STEAM 



353 



256.76 

"throttle 



limit, and a closer approach made to the Carnot cycle than in the ordinary en- 
gine. Thus, in Fig. 267, BCDS represents the Clausius cycle. The heat areas 
hi HE, gKJli, NMLg represent the withdrawal of steam from the 
various receivers, these amounts of heat being applied to heating 
the water along Bd, de, ef. The heat imparted from without is then 
only cfCDE. The work area DHIJKLMRS has been lost, but 
the much greater heat area ABfc has been saved, so that the effi- 
ciency is increased. The cycle is regenerative ; if the number of 
steps were infinite, the expansive path would be DF, parallel to 
BC, and the cycle would be equally efficient with that of Carnot. 
The actual efficiency was 68 per cent of that of the Carnot cycle. 
The steam rate was not low, being increased by the system of 
drawing off steam for the heaters from 11.4 to 12.26; but the real 
efficiency was, at the time, unsurpassed. A later test of a Nord- 
berg engine of similar construction, used to drive an air com- 
pressor, is reported by Hood (17). Here the combined diagrams 
were as in Fig. 268. Steam was received at 257 lb. pressure, the 
vacuum being rather poor. At normal capacity, 1000 hp., the 
mechanical efficiency was 90.35 per cent, and the heat consump- 
tion 169.29 B. t. u. This 
appears to be the best record 
to date. The efficiency is 
73.69 per cent of that of the 
Carnot cycle, and 88.2 per 
cent of that of the Clausius 
cycle. 




.24 CONDENSER 



Fig. 268. Art. 550. — Hood Compressor Diagrams. 



551. Superheated Steam; Reciprocating Engines. At 150 lb. pressure and 
250° of superheat, Schroter obtained heat rates of 199 to 223 B. t. u. with super- 
heated steam, against 213 to 246 B. t. u. with saturated steam in the same engine, 
the gain by superheating being greater at wide ranges of expansion. Jacobus 
(18) found on a small compound Rice and Sargent engine, a steam rate of 9.56 
lb. when about 400° of superheat was used, with a rate of 13.84 lb. for saturated 
steam. The pressure was 140 lb. and the vacuum only fair. The engine was, 
however, poorly adapted for the use of saturated steam. A result of exceptional 
interest was obtained in Carpenter's tests (19) of the engines of the White steam 
motor car. The maximum output w T as only 45 hp., the weight of the entire power 
plant only 643 lb. The engine was cross-compound, running condensing. The 
boiler pressure ranged up to 595 lb., with as much of 300° of superheat ; the 
exhaust from the engine was, in fact, superheated. A steam rate as low as 10.8 
lb. was obtained, or of 12 lb. per brake horse power, corresponding to 246 B. t. u. 
per brake horse power per minute. The Van den Kerchove engine mentioned in 
Art. 548 gave, with superheat, a steam rate of 8.99, and a heat unit consumption 
of 192 B. t. u. 



552. Turbines. With pressures of from 78.8 to 140 lb., and vacuum from 
24.3 to 26.4 in., steam rates per brake horse power of 18.0 to 23.2 lb. have been 



354 APPLIED THERMODYNAMICS 

obtained with saturated steam on De Laval turbines. Dean and Main (20) found 
corresponding rates of 15.17 to 16.54 with saturated steam at 200 lb. pressure, and 
13.94 to 15.62 with this steam superheated 91°. 

Parsons turbines, with saturated steam, have given rates per brake horse 
power from 14.1 to 18.2; with superheated steam, from 12.6 to 14.9. This was at 
120 lb. pressure. A 7500-kw. unit tested by Sparrow (21) with 177.5 lb. initial 
pressure, 95.74° of superheat, and 27 in. of vacuum, gave 15.15 lb. of steam per 
kw.-hr. The Stott engine-turbine outfit (see footnote 23, Chapter XIV) gave a 
thermal efficiency of 0.206 from steam to generator output while the load varied 
from 6500 kw. to 15,500 kw. Bell reports for the Lusitania (22) a coal consump- 
tion of 1.43 lb. per horse power delivered at the shaft. Denton quotes (23) 10.28 
lb. per brake horse power on a 4000 hp. unit, with 190° of superheat (214 B. t. u. per 
minute); and 13.08 on a 1500-hp. unit using saturated steam. A 400-kw. unit 
gave 11.2 lb. with 180° of superheat. A 1250- kw. turbine gave 13.5 lb. with 
saturated steam, 12.8 w T ith 100° of superheat, 13.25 with 77° of superheat (24). 
(All per brake hp.-hr.) 

A Rateau machine, with slight superheat, gave rates from 15.2 to 19.0 lb. per 
brake horse power. Curtis turbines have shown 14.8 to 18.5 lb. per kw.-hr., as the 
superheat decreased from 230° to zero, and of 17.8 to 22.3 lb. as the back pressure 
increased from 0.8 to 2.8 lb. absolute. Kruesi has claimed (25) for a 5000-kw. 
Curtis unit, with 125° of superheat, a steam rate of 14 lb. per kw.-hr. ; and for a 
2000-kw. unit, under similar conditions, 16.4 lb. 

553. Summary. The following table represents the best results as above 
given, with some of the results to be expected in ordinary practice with usual 
good engines operating at reasonably steady loads : 





SATUKATED STEAM 




Type of Engine 


Best Steam Rate 


Average Steam Rate 




IHP. 


IHP. 


Simple, Non-Condensing 


21.5 


38.0 


Compound, Non-Condensing 


19.14 


23.0 


Simple, Condensing 


16.5 


22.0 


Compound, Condensing 


11.22 


18.0 


Triple, Condensing 


11.05 





Quadruple, Condensing 


(169.29 B. t. u.) 

BHP. 




Single Stage Velocity Turbine 


15.17 





Pressure Turbine 


13.08 






SUPERHEATED STEAM 
IHP. 
Compound, Condensing 8.99 (192 B. t. u.) 

BHP. 

Single Stage Velocity Turbine • 13.94 
Pressure Turbine j imatel 10 

Multi-stage Velocity Turbine j 



ENGINE FRICTION 



355 



554. Locomotive Tests. The surprisingly low steam rate of 16.60 lb. has 
been obtained at 200 lb. pressure, with superheat up to 192°. This is equivalent 
to a rate of 17.8 lb. with saturated steam. The tests at the Louisiana Purchase 
Exposition (26) showed an average steam rate of 20.23 lb. for all classes of engines 
tested, or of 21.97 for simple engines and 18.55 for compounds, with steam pres- 
sures ranging from 200 to 225 lb. These results compare most favorably with any 
obtained from high-speed non-condensing stationary engines. The mechanical 
efficiency of the locomotive, in spite of its large number of journals, is high ; in 
the tests referred to, under good conditions, it averaged 88.3 per cent for consoli- 
dation engines and 89.1 per cent for the Atlantic type. The reason for these high 
efficiencies arises from the heavy overload carried in the cylinder in ordinary ser- 
vice. The maximum equivalent evaporation per square foot of heating surface 
varied from 8.55 to 16.34 lb. at full load, against a usual rate not exceeding 4.0 lb. 
in stationary boilers ; the boiler efficiency consequently was low, the equivalent 
evaporation per pound of dry coal (14,000 B. t. u.) falling from 11.73 as a maxi- 
mum to 6.73 as a minimum, between the extreme ranges of load. Notwithstand- 
ing this, a coal consumption of 2.27 lb. per Ihp.-hr. has been reached. These trials 
were, of course, laboratory tests; road tests, reported by Hitchcock (27), show less 
favorable results ; but the locomotive is nevertheless a highly economical engine, 
considering the conditions under which it runs. 



555. Engine Friction. Excepting in the case of turbines, the figures given 
refer usually to indicated horse power, or horse power developed by the steam in 
the cylinder. The effective horse power, exerted by the shaft, or brake horse 
power, is always less than this, by an amount depending upon the friction of the 
engine. The ratio of the latter to the former gives the mechanical efficiency, which 
may range from 0.85 to 0.90 in good practice with rotative engines of moderate 
size, and up to 0.965 in exceptional cases. The brake horse power is usually deter- 
mined by measuring the pull exerted on a friction brake applied to the belt wheel. 

When an engine drives a generator, the power indicated in the cylinder may be 
compared with that developed by the generator, and an over-all efficiency of 
mechanism thus obtained. The difficulties involved 
have led to the general custom, in turbine practice, of 
reporting steam rates per kw.-hr. Thurston has em- 
ployed the method of driving the engine as a ma- 
chine from some external motor, and measuring the 
power required by a transmission dynamometer. 

In direct-driven pumps, air compressors, and re- 
frigerating machines, the combined mechanical effi- 
ciency is found by comparing the indicator diagrams 
of the steam and pump cylinders. These efficiencies 
are high, on account of the decrease in number of 
bearings, crank pins, and crosshead pins. 

556. Variation in Friction. Theoretically, at FlG . m Art 556 ._ Engine 
least, the friction includes two parts : the initial Friction. 




356 



APPLIED THERMODYNAMICS 



r— 700- 
600- 

—500- 
400- 
















y 


























^ 


</ 






300- 








J- 










200 
— 100- 




£ 


s^/^ 




























1 


1 


5 2 


9 2 


> 3 


3 


) 40 



INDICATED HORSE POWER 



Fig. 270. Art. 556. — Wi Hans Line for Constant 
Initial Pressure. 



friction, that of the stuffing boxes, which remains practically constant ; and the 
load friction, of guides, pins, and bearings, which varies with the initial pressure 

and expansive ratio. By plotting 
concurrent values of the brake horse 
power and friction horse power, we 
thus obtain snch a diagram as that 
of Fig. 269, in which the height ah 
represents the constant initial fric- 
tion, and the variable ordinate xy 
the load friction, increasing in arith- 
metical proportion with the load. 
It has been found, however, that in 
practice the total friction is more 
affected by accidental variations in 
lubrication, etc., than by changes in 
load, and that it may be regarded as 
practically constant, for a given en- 
gine, at all loads. 
The total steam consumption of an engine at any load may then be regarded 
as made up of two parts : a constant amount, necessary to overcome friction ; and 
a variable amount, necessary to 
do external work, and varying 
with the amount of that work. 
Willans found that this latter 
part varied in exact arithmeti- 
cal proportion with the load, 
when the output of the engine 
was varied by changing the initial 
pressure; a condition repre- 
sented by the Willans line of 
Fig. 270 (28). The steam rate 
was thus the same for all loads, 
excepting as modified by fric- 
tion. (Theoretically, this 
should not hold, since lowering 
of the initial pressure lowers 
the efficiency.) When the load 
is changed by varying the ratio 

of expansion, the corrected steam rate tends to decrease with increasing ratios, 
and to increase on account of increased condensation ; there is, however, some 
gain up to a certain limit, and the Willans line must, therefore, be concave up- 
ward. Figure 271 shows the practically straight line obtained from a series of 
tests of a Parsons turbine. If the line for an ordinary engine were perfectly 
straight, with varying ratios of expansion, the indication would be that the gain 
by more complete expansion was exactly offset by the increase in cylinder con- 
densation. A jacketed engine, in which the influence of condensation is largely 
eliminated, should show a maximum curvature of the Willans line. 



-2400 














































































1600 










<&/ 
























# 


























*/ 


























-1200 


































800- 
-600- 




























































































10 20 


30 40 50 60 


70 80 


90 


loo no 120 



Fig. 271. 



ELECTRICAL HORSE POWER 

Art. 556, Prob. 16. — Willans Line for a 
Parsons Turbine. 



MECHANICAL EFFICIENCY 



357 



557. Variation in Mechanical Efficiency. With a constant friction loss, the 
mechanical efficiency must increase as the load increases, hence the desirability 
of running engines at full capacity. This is strikingly illustrated in the locomo- 
tive (Art. 554). Engines operating at serious variations in load, as in street rail- 
way power plants, may be quite wasteful on account of the low mean mechanical 
efficiency. A secondary effect enters here, on account of the rapidity of fluctuation 
of the load ; this leads to losses both mechanical and thermodynamic, which, 
although of importance, have never been satisfactorily analyzed. 



558. Limit of Expansion. Aside from cylinder condensation, engine friction 
imposes a limit to the desirable range of expansion. Thus, in Fig. 272, the line 
ab may be drawn such that the constant p 
pressure a represents that necessary to over- 
come the friction of the engine. If ex- 
pansion is carried below ab, say to c, the 
force exerted by the steam along be will be 
less than the resisting force of the engine, 
and will be without value. The maximum 
desirable expansion, irrespective of cylinder 
condensation, is reached at b. 



559. Distribution of Friction. Experi- « 
menting in the manner described in Art. 
555, Thurston ascertained the distribution 
of the friction load by successively rem ov- ^ 272 ^ 558 ._ Engine Friction 
ing various parts of the engine mechanism. and L i m i t of Expansion. 

Extended tests of this nature, made by 

Carpenter and Preston (29) indicate that from 35 to 47 per cent of the whole 
friction load is imposed by the shaft bearings, from 22 to 49 per cent by the piston, 
piston rod, pins, and slides (the greater part of this arising from the piston and 
rod), and the remaining load by the valve mechanism. 




(1) Trans. A. S. M. E., Proc. Inst. M. E., Zeits. Ver. Deutsch. Ing., etc. (See 
The Engineering Digest, November, 1908, p. 542.) (2) Proc. Inst. Mech. Eng., from 1889. 
(3) Engine Tests, by Geo. H. Barrus. (4) Steam Turbines, 1906, 208-267. (5) He- 
searches in Experimental Steam Engineering. (6) Peabody, Thermodynamics, 1907, 
244 ; White, Jour. Am. Soc. Nav. Engrs., X. (7) Trans. A. S. M. E., XXX, 6, 811. 
(8) Ewing, The Steam Engine, 1906, 177. (9) Denton, Tlie Stevens Institute Indi- 
cator, January, 1905. (10) Trans. A. S. M. E., XXIV, 1274. (11) Denton, op. cit. 
(12) Ewing, op. cit., 180. (13) Trans. A. S. M. E., XXI, 1018. (14) Ibid., XXI, 327. 
(15) Ibid., XXI, 793. (16) Ibid., XXI, 181. (17) Ibid., XXVIII, 2, 221. (18) Ibid., 
XXV, 264. (19) Ibid., XXVIII, 2, 225. (20) Thomas, Steam Turbines, 1906, 212. 
(21) Power, November, 1907, p. 772. (22) Proc. Inst. Nav. Archts., April 9, 1908. 
(23) Op. cit. (24) Trans. A. S. M. E., XXV, 745 et seq. (25) Power, December, 
1907. (26) Locomotive Tests and Exhibits, published by the Pennsylvania Railroad. 
(27) Trans. A. S. M. E., XXVI, 054. (28) Min. Proc. Inst. C. E., CXIV, 
(29) Peabody, op. cit., p. 296. 



358 APPLIED THERMODYNAMICS 

PEOBLEMS 

(See footnote, Art. 546.) 

1. Eind the efficiency ratio (thermal efficiency as compared with that of the 
Carnot cycle), for the best compound engine in Art. 548, if the vacuum was 27 in,,* 
the steam as received being dry. 

2. Eind the heat unit consumption of an engine using 30 lb. of dry steam per 
Ihp.-hr. at 100 lb. gauge pressure, discharging this steam at atmospheric pressure. How 
much of the heat (ignoring radiation losses) in each pound of steam is rejected ? 
What is the quality of the steam at release ? 

3. What is the mechanical efficiency of an engine developing 300 Ihp., if 30 hp. 
is employed in overcoming friction ? 

4. Why is it unprofitable to run multiple expansion engines non-condensing ? 

5. Check the heat unit consumption given for the Rice and Sargent engine in 
Art. 548, and find how much it increased at 20 per cent overload. 

6. Make deductions from Art. 549 as to the value of triple expansion and 
jacketing. 

7. Check all of the efficiency ratios given in Art. 550, assuming a vacuum of 26 
in. in each case. Explain the low heat unit consumption in spite of the high steam 
rate. 

8. Eind the heat unit consumption with superheat for the Rice and Sargent 
engine in Art. 551, if the vacuum was 27 in. 

9. What would have been the thermal efficiency of the White motor car engine 
in Art. 551 if the Carnot efficiency ratio had been equal to that of the Hood compressor 
(Art. 550) ? (The temperature of saturated steam at 595 lb. gauge pressure is 489° E.) 
Compare with the gas engine figures in Art. 343. 

10. Eind the heat unit consumptions corresponding to the figures in Art. 552 for 
De Laval turbines, assuming the vacuum to have been 27 in. 

11. Find the heat unit consumption for the 7500 kw. unit in Art. 552. 

12. Estimate the probable limit of boiler efficiency of the plant on the S.S. 
Lusitania, if the coal contained 14,200 B. t. u. per lb. 

13. Determine from data given in Art. 554 whether a coal consumption of 2.27 
lb. per Ihp.-hr is credible for a locomotive. 

14. Using values given for the coal consumption and mechanical efficiency, with 
how little coal (14,000 B. t. u. per pound), might a locomotive travel 100 miles at a speed 
of 50 miles per hour, while exerting a pull at the drawbar of 22,000 lb. ? Make compari- 
sons with Problem 8, Chapter II, and determine the possible efficiency from coal to 
drawbar. 

* Vacua are measured downward from atmospheric pressure. One atmosphere = 
14.696 lb. per square inch — — 29.921 inches of (mercury) vacuum. If p = absolute 
pressure, pounds per square inch, v = vacuum in inches of mercury, 



29.921 

14.696 
29.921 



14.696/ 
(29.921 - v). 



PROBLEMS 359 

15. An engine consumes 220 B. t. u. per Ihp.-min., 360 B. t. u. per kw.-min. of 
generator output. The generator efficiency is 0.93. What is the mechanical efficiency 
of the direct-connected unit ? 

16. From Fig. 271, plot a curve showing the variation in steam consumption per 
hp.-hr. as the load changes. 

17. An engine works between 150 and 2 lb. absolute pressure, the mechanical 
efficiency being 0.75. What is the desirable ratio of (hyberbolic) expansion, friction 
losses alone being considered ? 

18. If the mechanical efficiency of a rotative engine is 0.85, what should be its 
mechanical efficiency when directly driving an air compressor, based on the minimum 
values of Art. 559 ? 



CHAPTER XVI 

THE STEAM POWER PLANT 

560. Fuels. The complex details of steam plant management arise 
largely from differences in the physical and chemical constitution of 
fuels. " Hard " coal, for example, is compact and hard, while soft coal is 
friable ; the latter readily breaks up into small particles, while the former 
maintains its initial form unless subjected to great intensity of draft. 
Hard coal, therefore, requires more draft, and even then burns much less 
rapidly than soft coal ; and its low rate of combustion leads to important 
modifications in boiler design and operation in cases where it is to be used. 
Soft coal contains large quantities of volatile hydrocarbons ; these distill 
from the coal at low temperature, but will not remain ignited unless the 
temperature is kept high and an ample quantity of air is supplied. The 
smaller sizes of anthracite coal are now the cheapest of fuels, in propor- 
tion to their heating value, along the northern Atlantic seaboard ; but the 
supply is limited and the cost increasing. In large city plants, where 
fixed charges are high, soft coal is often commercially cheaper on account 
of its higher normal rate of combustion, and the consequently reduced 
amount of boiler surface necessary. The sacrifice of fuel economy in 
order to secure commercial economy with low load factors is strikingly 
exemplified in the "double grate" boilers of the Philadelphia Rapid 
Transit Company and the Interborough Rapid Transit Company of New 
York (1). 

561. Heating Value. The heating value of a fuel is determined by completely 
burning it in a calorimeter, and noting the rise in temperature of the calorimeter 
water. The result stated is the number of heat, units evolved per pound with 
products of combustion cooled down to 32° F. Fuel oil has a heating value 
upward of 18,000 B. t. u. per pound ; its price is too high, in most sections of the 
country, for it to compete with coal. Wood is in some sections available at low 
cost ; its heating value ranges from 6500 to 8500 B. t. u. The heating values of 
commercial coals range from 8800 to 15,000 B. t. u. Specially designed furnaces 
are usually necessary to burn wood economically. 

562. Boiler Room Engineering. While the limit of progress in steam engine 
economy has apparently been almost realized, large opportunities for improvement 
are offered in boiler operation. This is usually committed to cheap labor, with 

360 



EFFICIENCY OF COMBUSTION 



361 



insufficient supervision. Proper boiler operation can often cheapen power to a 
greater extent than can the substitution of a good engine for a poor one. New 
designs and new test records are not necessary. Efficiencies already reported 
equal any that can be expected; but the attainment of these efficiencies in ordi- 
nary operation is essential to the continuance in use of steam as a power produc- 
ing medium. 

563. Combustion. One pound of pure carbon burned in air uses 2.67 
lb. of oxygen, forming a gas consisting of 3.67 lb. of carbon dioxide and 
8.90 lb. of nitrogen. 
If insufficient air 
is supplied, the 
amount of carbon 
dioxide decreases, 
some carbon mon- 
oxide being 
formed. If the air 
supply is 50 per 
cent, deficient, no 
carbon dioxide can 
(theoretically at 
least) be formed. 
With air in excess, 
additional free 
oxygen and nitro- 
gen will be found 

in the products of combustion. Figure 273 illustrates the percentage 
composition by weight of the gases formed by combustion of pure carbon 
in varying amounts of air. The proportion of carbon dioxide reaches a 
maximum when the air supply is just right. 

























1 






























































£ 










































f 


Y 










































y 












































/ 




















































































5,000 £ 












/ 
































< 










/ 


/ 


\ 
































i0 








/ 






^ 


*fr 


























cc 








1 










\A 


kr 
























3000 Q 








A 












s 


k< 


% 




















Ol 




/ 


/ 




% 














■<. 


f**s 


:§£ 


!To 












2,000 £ 










V 
















<YG 


Tf 












\ 

1 








% 
















V^o 














CO 

1,000 * 


























PXlOE 

















i 






s 



























S 5 «o oo o *» ^ eOOOOfrJ-rOOOOOJ -»■ co So o 
AIR SUPPLY.»PERCENTAGE OF AM'T THEOR. NECESSARY FOR COMBUSTION 

Fig. 273. Arts. 563, 564. — Air Supply and Combustion. 



564. Temperature Rise. In burning to carbon dioxide, each pound of 
carbon evolves 14,500 B. t. u. In burning to carbon monoxide, only 4450 
B. t. u. are evolved per pound. Let W be the weight of gas formed per 
pound of carbon, K its mean specific heat, T — t the elevation of tempera- 



ture produced ; then for combustion to carbon dioxide, T — t 

4450 



14500 



and 



for combustion to carbon monoxide, T — t = 



WK 



WK 

The rise of tempera- 



ture is much less in the latter case. As air is supplied in excess, W 
increases while the other quantities on the right-hand sides of these equa- 
tions remain constant, so that the temperature rise similarly decreases. 
The temperature elevations are plotted in Fig. 273. The maximum rise 
of temperature occurs when the air supply is just the theoretical amount. 



362 APPLIED THERMODYNAMICS 

565. Practical Modifications. These curves truly represent the phe- 
nomena of combustion only when the reactions are perfect. In practice, 
the fuel and air are somewhat imperfectly mixed, so that we commonly 
find traces of free oxygen and carbon monoxide along with carbon dioxide. 
The best results are obtained by supplying some excess of air; instead of 
the theoretical 11.57 lb., about 16 lb. may be supplied, in good practice. 
In poorly operated plants, the air supply may easily run up to 50 or even 
100 lb., the percentage of carbon dioxide, of course, steadily decreasing. 
Gases containing 10 per cent, of dioxide by volume are usually considered 
to represent fair operation. 

566. Distribution of Heat. Of the heat supplied to the boiler by the fuel, 
ignoring radiation losses, a part is employed in making steam, a small amount of 
fuel is lost through the grate bars, some heat is transferred to the external atmos- 
phere, and some is carried away by the heated gases leaving the boiler. This 
last is the important item of loss. Its amount depends upon the weight of gases, 
their specific heat and temperature. The last factor we aim to fix in the design 
of the boiler to suit the specific rate of combustion : the specific heat we cannot 
control; but the weight of gas is determined solely by the supply of air, and is sub- 
ject to operating control. 

Efficient operation involves the minimum possible air supply in 
excess of the theoretical requirement; it is evidenced by the per- 
centage of carbon dioxide in the discharged gases. If the air supply 
be too much decreased, however, combustion may be incomplete, 
forming carbon monoxide, and another serious loss will be experienced, 
due to the potential heat carried off by the gas. 

567. Air Supply and Draft. The draft necessary is determined by the physical 
nature of the fuel ; the air supply, by its chemical composition. The two are not 
equivalent ; soft coal, for example, requires little draft, but ample air supply. The 
two should be subject to separate regulation. Low grade anthracite requires ample 
draft, but the air supply should be closely economized. If forced draft, by steam 
jet, blow 7 er, or exhauster, is employed, the necessary head should be provided with- 
out the delivery of an excessive quantity of air. 

568. Types of Boiler. Boilers are broadly grouped as fire-tube or water-tube, 
internally or externally fired. A type of externally fired w r ater-tube boiler has been 
shown in Fig. 233. In this, the Babcock and Wilcox design, the path of the gases 
is as described in Art. 508. The feed water enters the drum 6 at 29, flows down- 
ward through the back water legs at a, and then upward to the right along the 
tubes, the high temperature zone at 1 compelling the water above it in the tubes 
to rise. Figure 274 shows the horizontal tubular boiler, probably most generally 
used in this country. The fire grate is at S. The gases pass over the bridge wall 
O, under the shell of the boiler, up the back end F, and to the right through tubes 



STEAM BOILERS 



363 



running from end to end of the cylindrical shell. The tubes terminate at C, and 
the gases pass up and away. Feed water enters the front head, is carried in the 
pipe about two thirds of the distance to the back end, and then falls, a compensating 






-rjl 
i !! 



-zzl 



S 5 'I 

x.^ — _a — -i-lH-k 




upward current being generated over the grate. This is an externally fired fire-tube 
boiler. Figure 275 shows the well-known locomotive boiler, which is internally fired. 
The coldest part of this boiler is at the end farthest from the grate, on the exposed 
sides. The feed is consequently admitted here. Figure 276 shows a boiler com- 
monly used in marine service. The grate is placed in an internal furnace ; the 
gases pass upward in the back end, and return through the tubes. The feed pipe 
is located as in horizontal tubular boilers. 



364 



APPLIED THERMODYNAMICS 



569. Discussion. 




£ r 



The internally fired boiler requires no brick furnace 

setting, and is compact. 
The water-tube boiler is 
rather safer than the fire- 
tube, and requires less 
space. It can be more 
readily used with high 
steam pressures. The im- 
portant points to observe 
in boiler types are the 
paths of the gases and of 
the water. The gases 
should, for economy, im- 
pinge upon and thoroughly 
circulate about all parts 
of the heating surface; 
the circulation of the 
water for safety and large 
capacity should be posi- 
tive and rapid, and the 
cold feed water should be 
introduced at such a point 
as to assist this circula- 
tion. 

There is no such thing 
as a "most economical 
type" of boiler. Any 
type may be economical 
if the proportions are 
right. The grade of fuel 
used and the draft attain- 
able determine the neces- 
sary area of grate for a 
given fuel consumption. 
The heating surface must 
be sufficient to absorb the 
heat liberated by the fuel. 
The higher the rate of 
combustion (pounds of fuel 
burned per square foot of 
grate per hour), the greater 
the relative amount of 
heating surface necessary. 



STEAM BOILER ECONOMY 



365 




L.ON.GJTUDIISAL SECTION 

Fig. 270. Art. 568. — Marine Boiler. (The Bigelow Company.) 

Bates of combustion range from 12 lb. with low grade hard coal and 
natural draft up to 30 or 40 lb. with soft coal ; # the corresponding ratios 
of heating surface to grate surface may vary from 30 up to 60 or 70. 
The best economy has usually been associated with high ratios. The 
rate of evaporation is the number of pounds of water evaporated per 
square foot of heating surface per hour; it ranges from 3.0 upward, de- 
pending upon the activity of circulation of water and gases. An effective 
heating surface usually leads to a low flue-gas temperature and relatively 
small loss to the stack. Small tubes increase the efficiency of the heat- 
ing surface but may be objectionable with certain fuels. Tubes seldom 
exceed 20 ft. in length. In water-tube boilers, the arrangement of tubes 
is important. If the bank of tubes is comparatively wide and shallow, 
the gases may pass off without giving up the proper proportion of their 
heat. If the bank is made too high and narrow, the grate area may be 
too much restricted. The gases must not be allowed to reach the flue too 
quickly. 

570. Boiler Capacity. A boiler evaporating 3450 lb. of water per hour 
from and at 212° F. performs 970 x 778 x 3450 = 2,600,000,000 foot-pounds 



* Much higher rates are attained in locomotive practice ; and in torpedo boats, with 
intense draft, as much as 200 lb. of coal may be burned per square foot of grate per hour. 



366 APPLIED THERMODYNAMICS 

of work, or 1300 horse power. Xo engine can develop this amount of power 
from 3450 lb. of steam per hour ; the power developed by the engine is 
very much less than that by the boiler which supplies it. Hence the custom 
of rating boilers arbitrarily. By definition of the American Society of 
Mechanical Engineers, a boiler horse power means the evaporation of 341 lb. 
of water per hour from and at 212° F. This rating was based on the 
assumption (true, in 1876, when the original definition was established) 
that an ordinary good engine required about this amount of steam per 
horse power hour. This evaporation involves the liberation of about 
33,000 B. t. u. per hour. 

571. Limit of Efficiency. The gases cannot leave the boiler at a 
lower temperature than that of the steam in the boiler. Let t be the 
initial temperature of the fuel and air, x the temperature of the steam, 
and T the temperature attained by combustion ; then if W be the 
weight of gas and K its specific heat, assumed constant, the total 
heat generated is WK(T — £), the maximum that can be utilized is 
WK(T — x), and the limit of efficiency is 

T-x 
T-i 

In practice, we have as usual limiting values jP=4850, ^=350, t = 60; 
whence the efficiency is 0.94 — a value never reached in practice. 

572. Boiler Trials. A standard code for conducting boiler trials has 
been published by the American Society of Mechanical Engineers (2). 
A boiler, like any mechanical device, should be judged by the ratio of the 
work which it does to the energy it uses. This involves measuring the 
fuel supplied, determining its heating value, measuring the water evaporated, 
and the pressure, superheat, or wetness of the steam. The result is usually 
expressed in pounds of dry steam evaporated per pound of coal from and at 
212° F., briefly called the equivalent evaporation. 

Let the factor of evaporation be F. If W pounds of water are fed to 
the boiler per pound of coal burned, the equivalent evaporation is FW. If 
C be the heating value per pound of fuel, the efficiency is 970 FW-i- O. 
Many excessively high values for efficiency have been reported in conse- 
quence of not correcting for wetness of the steam ; the proportion of wet- 
ness may range up to 4 per cent, in overloaded boilers. The highest well- 
confirmed figures give boiler efficiencies of about 83 per cent. The average 
efficiency, considering all plants, probably ranges from 0.40 to 0.60. 

573. Checks on Operation. A careful boiler trial is rather expensive, and 
must often interfere with the operation of the plant. The best indication of cur- 



CHIMNEY DESIGN 367 

rent efficiency obtainable is that afforded by analysis of the flue gases. It has 
been shown that maximum efficiency is attained when the percentage of carbon 
dioxide reaches a maximum. Automatic instruments are in use for continuously 
determining and recording the proportion of this constituent present in flue gases. 

574. Boiler and Furnace Efficiency. This measurement (Art. 573) in reality 
indicates principally the furnace efficiency, which may be defined as the quotient 
of the available heat (above the temperature of the steam) in the gases, per pound 
of fuel supplied, by the heat in a pound of fuel. The boiler surface efficiency, sepa- 
rately considered, is then the quotient of the heat taken up by the steam, by the 
heat originally available in the gases. It can be estimated by noting the tempera- 
ture of the escaping flue gases. In trials, it is rarely possible to accurately distin- 
guish between the two efficiencies. 

575. Chimney Draft. In most cases, the high temperature of the flue gases 
leaving the boiler is utilized to produce a natural upward draft for the mainte- 
nance of combustion. At equal temperatures, the chimney gas w T ould be heavier 
than the external air in the ratio (n + 1)h- n, where n is the number of pounds of 
air supplied per pound of fuel. If T, t denote the respective absolute tempera- 
tures, then, the density of the outside air being 1, that of the chimney gas is 

At 60° F., the volume of a pound of air is 13 cu. ft. The weight of 



A n I 



gas in a chimney of cross-sectional area A and height H is then 

AH T(n±l\ 13< 



The " pressure head," or draft, due to the difference in weight inside and outside 
is, per unit area, 

[ t\ n 1 J 

This is in pounds per square foot, if appropriate units are used ; drafts are, how- 
ever, usually stated in " inches of water," one of which is equal to 5.2 lb. per square 
foot. The force of draft therefore depends directly on the height of the chimney ; 
and since n + 1 is substantially equal to n, maximum draft is obtained when T-± t 
is a minimum, or (since T is fixed) when t is a maximum; in the actual case, 
however, the quantity of gas passing would be seriously reduced if the value of t 
"were too high, and best results (3), so far as draft is concerned, are obtained when 
t: T::25:12. 

To determine the area of chimney : the velocity of the gases is, in feet per 
second, 

v = VTgrh = 8.03 Vh = 8.03\|' 

h being the head corresponding to the pressure p and density d of the gases in the 
chimney. Also 

13(1 n / 



368 APPLIED THERMODYNAMICS 

Then if C lb. of coal are to be burned per hour, the weight of gases per second is 

££±11, their volume is ^L+D, 
3600 3600 d 

and the area of the chimney, in square feet, is 

3600 d V d 

A slight increase may be made to allow for decrease of velocity at the sides. Most 
chimney tables are based on an air supply of about 75 lb. per pound of fuel 
(Art. 565). 

576. Mechanical Draft. In lieu of a chimney, steam-jet blowers or fans may 
be employed. These usually cost less initially, and more in maintenance. The 
ordinary steam-jet blower is wasteful, but the draft is independent of weather con- 
ditions, and may be greatly augmented in case of overload. The velocity of the 

air moved by a fan is 

v = v2 gh, 

where h is the head due to the velocity, equal to the pressure divided by the 

density. Then / ■ 2 , 

v=\'2g'- and j p = — . 
* d 2g 

If a be the area over which the discharge pressure p is maintained, the work 
necessary is W = pav = dav^ 2 g. 

We may note, then, that the velocity of the air and the amount delivered 
vary as the peripheral speed of the wheel, its pressure as the square, and the 
power consumed as the cube, of that speed. Low peripheral speeds are 
therefore economical. They are usually fixed by the pressure required, 
the fan width being then made suitable to deliver the required volume. 

577. Forms of Fan Draft. The air may be blown into a closed fire room or 
ash pit or the flue gases may be sucked out by an induced draft fan. In the latter 
case, the high temperature of the gases reduces the capacity of the fan by about 
one half ; i.e. only one half the weight of gas will be discharged that would be 
delivered at 60° F. Since the density is inversely proportional to the absolute 
temperature, the required pressure can then be maintained only at a considerable 
increase in peripheral speed; which is not, however, accompanied by a concordant 
increase in power consumption. Induced draft requires the handling of a greater 
weight, as well as of a greater volume of gas, than forced draft ; the necessary 
pressure is somewhat greater, on account of the frictional resistance of the flues 
and passages; high temperatures lead to mechanical difficulties with the fans. 
The difficulty of regulating forced draft has nevertheless led to a considerable 
application of the induced system. 

578. Stokers. Mechanical stokers are often used when soft coal is employed 
as fuel. Besides saving some labor, in large plants at least, they give more per- 
fect combustion of hydrocarbons, with reduced smoke production. Figure 277 



SUPERHEATERS 



369 



shows, incidentally, a modern form of the old " Dutch oven " principle for soft 
coal firing. The flames are kept hot, because they do not strike the relatively cold 
boiler surface until combustion is complete. Fuel is fed alternately to the two 
sides of the grate, ro that the smoking gases from one side meet the hot flame 
from the other at the hot baffling " wing walls " a, b. The principle involved in 




Fig. 277. Arts. 578, 579. — Sectional Elevation of Foster Superheater combined with Boiler 
and Kent Wing Wall Furnace. (Power Specialty Company.) 



wm 



WEJM^m. 




Fig. 278. Arts. 578, 579. — Babcock and Wilcox Boiler with Chain Grate Stoker and 

Superheater. 



370 



APPLIED THERMODYNAMICS 



the attempt to abate smoke is that of all mechanical stokers, which may be grouped 
into three general types. In the chain grate, coal is carried forward continuously 
on a moving chain, the ashes being dropped at the back end. The gases from 
the fresh fuel pass over the hotter coke fire on the back portion of the grate. (See 
Fig. 278.) The second type comprises the inclined grate stokers. The high com- 
bustion chamber above the lower end of the grate is a decided advantage with 
many types of boilers. The smoke is distilled off at the "coking plate." The 
underfeed stoker feeds the coal by means of a worm to the under side of the fire, 
and the smoke passes through the incandescent fuel. All stokers have the ad- 
vantage of making firing continuous, avoiding the chilling effect of an open fire 
door. 

579. Superheaters ; Types. Superheating was proposed at an early date, and 
given a decided impetus by Hirn. After 1870, as higher steam pressures were 
introduced, superheating was partially abandoned. Lately, it has been reintro- 




FlG. 279. Art. 579. — Cole Superheater. (American Locomotive Company.) 



duced, and the use of superheat is now standard practice in France and Germany, 
while being quite widely approved in this country. Superheaters may be sepa- 
rately fired, steam from a boiler being passed through an entirely separate machine, 
or, as is more common, steam may be carried away from the water to some space 
provided for it within the boiler setting or flue, and there heated by the partially 
spent gases. When it is merely desired to dry the steam, the " superheater " may 
be located in the flue, using waste heat only. When any considerable increase 
of temperature is desired, the superheater should be placed in a zone of the 
furnace where the temperature is not less than 1000° F. With a difference in 



FEED WATER HEATERS 



371 



mean temperature between gases and steam of 400° F., about 5 B. t. u. may be 
transmitted per degree of mean temperature difference per square foot of surface 
per hour (4). The location of the Babcock and Wilcox superheater is shown in 
Fig. 277 ; a similar arrangement, in 
which the chain grate stoker is inci- 
dentally represented, is shown in Fig. 
278. In locomotive service, Field tubes 
may be employed, as in Fig. 279, the 
steam emerging from the boiler at a, 
and passing through the header b to the 
small tubes c, c, c, in the fire tubes d, d, 
rf(5). 

A typical superheater tube or "ele- 
ment" is shown in Fig. 280. This is 
made double, the steam passing through 
the annular space. Increased heating 
surface is afforded by the cast iron rings 
a, a. In some single-tube elements, the 
heating surface is augmented by internal 
longitudinal ribs. The tubes should be located so that the wettest steam will 
meet the hottest gases. 




Fig. 



280. Art. 579. — Superheater Element. 
(Power Specialty Company.) 



580. Feed-water Heaters. In Fig. 233, the condensed water is returned 
directly from the hot well 24, by way of the feed pump IV, to the boiler. This 
water is seldom higher in temperature than 125° F. A considerable saving may 
be effected by using exhaust steam to further heat the water before it is delivered 
to the boiler. The device for accomplishing this is called the feed-water heater. 
With a condensing engine, as shown, the water supply may be drawn from the 
hot well and the necessary exhaust steam supplied by the auxiliary exhausts 27 
and 31 ; 1 lb. of steam at atmospheric pressure should .heat about 9.7 lb. of 
water through 100°. Accurately, W(xL — <7 )= iv(Q — q), in which W is the 
weight of steam condensed, x is its dryness, L its latent heat, and w is the weight 
of feed water, the initial and final heat contents of which are respectively q and Q. 
The heat contents of the steam after condensation are q . Then 

Q-q 
With non-condensing engines, the exhaust steam from the engines themselves 
is used to heat the cold incoming water. 



581. Types. Feed-water heaters may be either "open," the steam and water 
mixing, or " closed," the heat being transmitted through the surface of straight 
or curved tubes, through which the water circulates. Figure 281 shows a closed 
heater; steam enters at A and emerges at B; water enters at C, passes through 
the tubes and out at D. The openings E, E are for drawing off condensed steam. 
An open heater is shown in Fig. 282. Water enters through the automatically 
controlled valve a, steam enters at b. The water drips over the trays, becoming 
finely divided and effectively heated by the steam. At c there is provided storage 



372 



APPLIED THERMODYNAMICS 



space for the mixture, and at d is a bed of coke or other absorbent material, 
through which the water niters upward, passing out at e. The open heater usu- 
ally makes the water rather hotter, and lends itself more readily to the reclaiming 



B 





Fig. 281. Art. 581. — Wheeler Feed Water Heater. 



of hot drips from the steam pipes, returns from heating systems, etc., than a 
heater of the closed type. Live steam is sometimes used for feed-water heating ; 
the greater effectiveness of the boiler heating surface claimed to arise from intro- 
ducing the water at high temperature has been disputed (6) ; but the high tem- 
peratures possible with live steam are of decided 
value in removing dissolved solids, and the waste of 
steam may be only slight. Closed heaters are, 
of course, used for this service, as also with the 
isodiabatic multiple-expansion cycle described in 
Art. 550. Removal of some of the suspended and 
dissolved solids is also possible in ordinary open 
exhaust-steam heaters. Various forms of feed- 
water niters are used, with or without heaters. 

582. The Economizer. This is a feed-water 
heater in which the heating medium is the waste 
gas discharged from the boiler furnace. It may 
increase the feed temperature to 300° F. or more, 
whereas no ordinary exhaust-steam heater can pro- 
duce a temperature higher than 212° F. The gain 
by heating feed water is about 1 B. t. u. per pound 
of steam for each degree heated; or since average 
steam contains 1000 B. t, u. net, it is about 1 per cent for each 10° that the tem- 
perature is raised; precisely, the gain is (H - h) -*- Q, in which Q is the total heat 
of the steam gained from the temperature of feed to the state at evaporation and 
h and H the total heats in the water before and after heating. If T, t be the 
temperatures of flue gases and steam, respectively, IF the weight, and K the mean 
specific heat of the gases (say about 0.21), then the maximum saving that can be 
effected by a perfect economizer is WK(T - t). Good operation decreases W and 
T and thus makes the possible saving small. A typical economizer installation 




Fig. 282. 



Open Feed 



Art. 581.- 
Heater. 
(Harrison Safety Boiler Works.) 



THE ECONOMIZER 



373 



is shown in Fig. 283 ; arrangement is always made for by-passing the gases, as 
shown, to permit of inspecting and cleaning. The device consists of vertical cast- 
iron tubes with connecting headers at the ends, the tubes being sometimes stag- 
gered so that the gases will im- 
pinge against them. The external 
surface of the tubes is kept clean 
by scrapers, operated from a small 
steam engine. The tubes obstruct 
the draft, and some form of me- 
chanical draft is employed in con- 
junction with economizers. About 
4.8 sq. ft. of economizer surface 
are ordinarily used per boiler horse 
power. 

583. Miscellaneous Devices. 
A steam separator is usually placed 
on the steam pipe near the engine. 
This catches and more or less 
thoroughly removes any condensed 
steam, which might otherwise 
cause damage to the cylinder. 
Steam meters are being introduced 
for approximately indicating the 
amount of steam flowing through 
a pipe. Some of them record their 
indications on a chart. Feed-water 
measuring tanks are sometimes in- 
stalled, where periodical boiler 
trials are a part of the regular 
routine. The steam loop is a de- 
vice for returning condensed steam 
direct to the boiler. The drips are 
piped up to a convenient height, 
and the down pipe then forms a 
radiating coil, in which a consid- 
erable amount of condensation oc- 
curs. The weight of this column 
of water in the down pipe offsets a 
corresponding difference in pres- 
sure, and permits the return of 
drips to the boiler even when their 
pressure is less than the boiler 
pressure. The ordinary steam trap 
merely removes condensed water 
without permitting the discharge of uncondensed steam. Oil separators are some- 
times used on exhaust pipes to keep back any traces of cylinder oil. 




374 



APPLIED THERMODYNAMICS 




Fig. 284. Art. 584. — Sav- 
ing Due to Condensation. 



584. Condensers. The theoretical gain by running condensing is shown by 
the Carnot formula (T — t) -f- T. The gain in practice may be indicated on the 
PV diagram, as in Fig. 284. The shaded area represents 
work gained due to condensation ; it may amount to 10 
or 12 lb. of mean effective pressure, which means about 
a 25 per cent gain, in the case of an ordinary simple 
engine. This gain is principally the result of the in- 
troduction of cooling water, which usually costs merely 
the power to pump it ; in most cases, some additional 
power is needed to drive an air pump as well. 

In the surface condenser the steam and the water do 
not come into contact, so that impure water maybe used, 
as at sea, even when the condensed steam is returned to 
the boilers. The amount of condensing surface is 
usually computed from Whitham's empirical formula (7) S = WL 4- 180(r — t), 
in which S is in square feet, W is the weight of steam condensed per hour, L the 
latent heat at the temperature T of the steam, and t is the mean temperature of 
the circulating water between inlet and outlet. Let u, U be the 
initial and final temperatures of the water ; then the weight w of 
water required per hour is WL -^{U — u). The weight of water 
is often permitted to be about 40 times the weight of steam, a 
considerable excess being desirable. The outlet temperature of the 
water in ordinary surface condensers may be from 15° to 40° below 
that of the steam. 

The jet condenser is shown in Fig. 285. The steam and water 
mix in a chamber above the air pump cylinder, and this cylinder is 
utilized to draw in the water, if the lift is not excessive. Here, 
theoretically, U = T\ the supply of water necessary is less than in 
surface condensers. The boilers may be fed from the hot well, 
as in Fig. 233 (which shows a jet 
condenser installation), only when 
the condensing water is pure. 

The siphon condenser is shown 
in Fig. 286. Condensation occurs 
in the nozzle a, and the fall of 
water through b produces the 
vacuum. To '"preserve this, the 
lower end of the discharge pipe 
must be sealed as shown. The 
vacuum would draw water up the 

pipe b and permit it to flow over into the engine, if it were not that the length cd 
is made 31 ft. or more, thus giving a height to which the atmospheric pressure 
cannot force the water. Excellent results have been obtained with these con- 
densers without vacuum pumps. In some cases, however, a " dry " vacuum pump 
is used to remove air and vapor from above the nozzle. The vacuum will lift 
the inlet water about 20 ft., so that, unless the suction head is greater than this, no 
water supply pump is required, after the condenser is started. 




Fig. 285. 



Art. 584. — Horizontal Independent Jet 
Condenser. 



CONDENSERS 



375 



585. Evaporative Condensers. Steam has occasionally been condensed by 
allowing it to pass through coils over which fine streams of water trickled. The 



fe/tefVafre 



fc 



*Z-t 



I 



1 



[li/Wi 



ine 







% 



^ 



evaporation of the water (which 
may be hastened by a fan) cools 
the coils and condenses the steam, 
which is drawn off by an air 
pump. With ordinary condensers 
and a limited water supply cooling 
towers are sometimes used. These 
may be identical in construction 
with the evaporative condensers, 
excepting that warm water enters 
the coils instead of steam, to be 
cooled and used over again; or 
they may consist of open wood 
mats over which the water falls 
as in the open type of feed-water 
heater. Evaporation of a portion 
of the water in question (which 
need not be a large proportion of 
the whole) and warming of the 
air then cools the remainder of 
the water, the cooling being facili- 
tated by placing the mats in a 

.. , / , x & Al , . . , Fig. 286. Art. 581. — Bulkley Injector Condenser, 

cylindrical tower through which 

there is a rapid upward current of air, naturally or artificially produced (8). 



1 



c 

-r 

i 

i 

r 
I 
\ 



i 

-* 

t 
I 
i 
l 
i 
I 



586. Boiler Feed Pump. This may be either steam-driven or power-driven 
(as may also be the condenser pumps). Steam-driven pumps should be of the 
duplex type, with plungers packed from the outside, and with individually acces- 
sible valves. If they are to pump hot water, special materials must be used for 
exposed parts. The power pump has usually three single-acting water cylinders. 
There is much discussion at the present time as to the comparative economy of 
steam- and power-driven auxiliaries. The steam engine portion of an ordinary 
small pump is extremely inefficient, while power-driven pumps can be operated, at 
little loss, from the main engines. The general use of exhaust steam from aux- 
iliaries for feed-water heating ceases to be an argument in their favor when econo- 
mizers are used ; and in large plants the difference in cost of attendance in favor 
of motor-driven auxiliaries is a serious item. 



587. The Injector. The pump is the standard device for feeding stationary 
boilers; the injector, invented by Giffard about 1858, is used chiefly as an auxil- 
iary, although still in general application as the prime feeder on locomotives. It 
consists essentially of a steam nozzle, a combining chamber, and a delivery tube. 
In Fig. 287, steam enters at A and expands through B, the amount of expansion 
being regulated by the valve C. The water enters at D, and condenses the 
steam in E. We have here a rapid adiabatic expansion, as in the turbine ; the 



376 



APPLIED THERMODYNAMICS 



velocity of the water is augmented by the impact of the steam, and is in turn con- 
verted into pressure at F. Jn starting the injector, the water is allowed to flow 
away through G ; as soon as the velocity is sufficient, this overflow closes. An in- 
jector of this form will lift the water from a reasonably low suction level ; when 
the water flows to the device by gravity, the valve C may be omitted. 




MnTMTlMHrtfflM 



F m-+/o, 

UllilllllllllllWHUICOa 



To. boiler 



Fig. 287. Art. 587. — Injector. 



A self-starting injector is one in which the adjustment of the overflow at G is 
automatic. The ejector is a similar device by which the lifting of water from 
a well or pit against a moderate delivery head (or none) is accomplished. The 
siphon condenser (Art. 584) involves an application of the injector principle. The 
double injector is a series of successive injectors, one discharging into another. 

588. Theory. Tet x, L, h be the state of the steam, H the heat in the 
water, and v its velocity ; Q the heat in the discharged water at its veloc- 
ity V. The heat in one pound of steam is xL -f- h ; the heat in one pound 
of water supplied is H, and its kinetic energy v 2 -r- 2g\ the heat in one 
pound of discharge is Q, and its kinetic energy V 2 -f- 2 g. Let each pound 
of steam draw in y pounds of water; then 



xL + h + y [H+ 



V 2 



2g) 



(i + y)(Q + 



29 



The values of — and — - may ordinarily be neglected, and 



2g 2g 



y = 



xL + h—Q 
Q-H 



THE INJECTOR 377 

In another form, y(Q — H)=xL + h— Q, or the heat gained by the water 
equals that lost by the steam. This, while not rigidly correct, on account 
of the changes in kinetic energy, is still so nearly true that the thermal 
efficiency of the injector may be regarded as 100 per cent ; from this stand- 
point, it is merely a live-steam feed-water heater. 

589. Application. The formula given shows at once the relation between 
steam state, water temperature, and quantity of water per pound of steam. As 
the water becomes initially hotter, less steam is required ; but injectors do not 
handle hot water well. Exhaust steam may be used in an injector : the pressure 
of discharge is determined by the velocity induced, and not at all by the initial 
pressure of the steam; a large steam nozzle is required, and the exhaust injector 
will not ordinarily lift its own water supply. 

590. Efficiency. Let S be the head against which discharge is made ; 
then the work done per pound of steam is (1 -f y) S foot-pounds ; the 
efficiency is S(l + .y)-j- (xL + h— Q), ordinarily less than one per cent. 

This is of small consequence, as practically all of the heat not changed to 
work is returned>*to the boiler. Let W be the velocity of the steam issuing from 
the nozzle; then, since the momentum of a system of elastic bodies remains con- 
stant during impact, W + yv —(I + y) V. The value of W may be expressed in 
terms of the heat quantities by combining this equation with that in Art. 588. The 
other velocities are so related to each other as to give orifices of reasonable size. 
The practical proportioning of injectors has been treated by Kneass (9). 

(1) Finlay, Proc. A. L E. E., 1907. (2) Trans. A. S. M. E., XXI, 34. (3) Ran- 
kine, The Steam Engine, 1897, 289. (4) Longridge. Proc. Inst. M. E., 1896, 175. 
(5) Trans. A. S. M. E., XXVIII, 10, 1606. (6) Bilbrough, Power, May 12, 1908, 
p. 729. (7) Trans. A. S. 31. E., IX, 431. (8) Bibbins, Trans. A. S. 31. E., XXI, 11. 
(9) Practice and Theory of the Injector. 

SYNOPSIS OF CHAPTER XVI 

Hard coal requires high draft : soft coal, a high rate of air supply. 

In spite of its higher cost, commercial factors sometimes make soft coal the cheaper 
fuel. 

Heating values : fuel oil, 18,000; wood, 6500-8500; coals, 8800-15,000; B. t. u. per lb. 

The proportion of carbon dioxide in the flue gases reaches a maximum when the air 
supply is just right ; this is also the condition of maximum temperature and theo- 
retical efficiency. 

Advance in steam power economy is a matter of regulation of air supply ; economy 
may be indicated by automatic records of carbon dioxide. 

Types of boiler : water-tube, horizontal tubular, locomotive, marine : conditions of 
efficiency. 

Attention should be given to the circulation of the gases and the water. 

A boiler hp. = 34| lb. of water per hour from and at 212° F.: approximately 33,000 
B. t. u. per hour. 



378 APPLIED THERMODYNAMICS 

T— x 
Limit of efficiency = ; say 0.94 ; never reached in practice. 

lip&t in stp^m 
Boiler efficiency = — — — : — ^ — r ; usually 0.40 to 0.60 ; may be 0.83. 
iieat in iuel 

Furnace efficiency = heat in S ases . Heating surface efficiency = heat in steam . 
heat in fuel heat in gases 

Chimney draft = h\i- — (*L±1) I -*- 13 : area = C ( M + 1 ) -^ 8.03 A /P . 

Fan draft : v— V'2 gh, p = — , W — — — : slow speeds advantageous. 
2g 2g 

In induced draft, the fan is between the furnace and the chimney ; in forced draft, it 

delivers air to the ash pit. 
Mechanical stokers (inclined grate, chain grate, underfeed), used with soft coal, aim 

to give space for the hydrocarbonaceous flame without permitting it to impinge on 

cold surfaces. 
Superheaters may be located in the flue, or, if much superheating is required, may be 

separately fired. About 5 B. t. u. may be the transmission rate. 

Feed-water heaters may be open or closed: w = — ^ — ~ ^ ; for open heaters, q = Q. 

Q-Q 
The economizer uses the waste heat of the flue gases : saving per pound of fuel 

= WK(T-t). 
Condensers may be surface, jet, evaporative, or siphon : w = WL -f- ( U — u) ; 

S = WL tt- 180( T — t). The siphon condenser may operate without a vacuum 

pump. 
The use of steam-driven auxiliaries affords exhaust steam for feed-water heating. 
The injector converts heat energy into velocity: y = - — l ~ ^ ; efficiency = 

^±vl_ w+yv ^ + y) v. q ~ H 

xL-r h— Q 

•i 

PROBLEMS 

1. One pound of pure carbon is burned in 16 lb. of air. Assuming reactions to be 
perfect, find the percentage composition of the flue gases and the rise in temperature, 
the specific heats being, C0 2 , 0.215 ; N, 0.245 j O, 0.217. 

2. A boiler evaporates 3000 lb. of water per hour from a feed-water temperature 
of 200° E. to dry steam at 160 lb. pressure. What is its horse power ? 

3. In Problem 2, what proportion of the whole heat in the fuel is carried away 
in the flue gases, if their temperature is 600° F., assuming the specific heats of the 
gases to be constant ? The initial temperature of the fuel and air supplied is 0° F. 

4. The boiler in Problem 2 burns 350 lb. of coal (14,000 B. t. u. per pound) per hour. 
What is its efficiency ? 

5. In Problems 1 and 3, the temperature of the steam is 350° F. Find the furnace 
efficiency and the efficiency "of the heating surface (Art. 574). 

6. In Problem 1, if the gas temperature is 600° F., the air temperature 60° F., 
compare the densities of the gases and- the external air. What must be the height of a 
chimney to produce, under these conditions, a draft of 1 in. of water? Find the 
diameter of the (round) chimney to burn 5000 lb. of coal per hour. 



THE STEAM POWER PLANT 379 

7. Two fans are offered for providing draft in a power plant, 15,000 cu. ft. of 
air being required at \\ oz. pressure per minute. The first fan has a wheel 30 in. in 
diameter, exerts 1 oz. pressure at 740 r. p. m., delivers 405 cu. ft. per minute, and con- 
sumes 0.16 hp., both per inch of wheel width and at the given speed. The second fan 
has a 54-inch wheel, runs at 410 r. p. m. when exerting 1 oz. pressure, and delivers 
726 cu. ft. per minute with 0.29 hp., both per inch of wheel width and at the given 
speed. Compare the widths, speeds, peripheral speeds, and power consumptions of the 
two fans under the required conditions. 

8. Dry steam at 350° F. is superheated to 450° F. at 135 lb. pressure. The flue 
gases cool from 900° F. to 700° F. Find the amount of superheating surface to provide 
for 3000 lb. of steam per hour, and the weight of gas passing the superheater. If 280 
lb. «of coal are burned per hour, what is the air supply per pound of coal ? 

9. Water is to be raised from 60° F. to 200° F. in a feed-water heater, the weight 
of water being 10,000 lb. per hour. Heat is supplied by steam at atmospheric pressure, 
0.95 dry. Find the weight of stsam condensed («) in an open heater, (&) in a closed 
heater. Find the surface necessary in the latter (Art. 584) . 

10. In Problem 3, what would be the percentage of saving due to an economizer 
which reduced the gas temperature to 400° F. ? 

11. An engine discharges 10,000 lb. per hour of steam at 2 lb. absolute pressure, 
0.95 dry.. Water is available at 60° F. Find the amount of water supplied for a jet 
condenser. Find the amount of surface, and the water supply, for a surface condenser 
in which the outlet temperature of the water is 85° F. If the surface condenser is 
operated with a cooling tower, what weight of water will theoretically be evaporated in 
the tower, assuming the entire cooling to be due to such evaporation. (N.B. A large 
part of the cooling is in practice effected by the air.) 

12. Find the dimensions of the cylinders of a triplex single-acting feed pump to 
deliver 100,000 lb. of water per hour at 60° F. at a piston speed of 100 ft. per minute 
and 30 r. p. m. 

13. Dry steam at 100 lb. pressure delivers 3000 lb. of water per hour from an injec- 
tor at 165° F., the inlet temperature of the water being 60° F. Find the weight of 
steam used. The water is measured on the inlet side of the injector. 

14. In Problem 13, the boiler pressure is 100 lb. What is the efficiency of the 
injector, considered- as a pump ? 

15. In Problem 13, the velocity of the entering water is 12 ft. per second, that of the 
discharge is 1 14 ft. per second. Find the velocity of the steam leaving the discharge 
nozzle. 

16. What is the relation of altitude to chimney draft ? (See Problem 12, Chapter 
XIII.) 



CHAPTER XVII 

DISTILLATION — FUSION— LIQUEFACTION OF GASES 
Vacuum Distillation 




WATER OUTLET/ 



Fig. 288. Art. 591. —Still. 



591. The Still. Figure 288 represents an ordinary still, as used for 
purifying liquids or for the recovery of solids in solution by concentration. 
Externally applied heat evaporates the liquid in A, which is condensed at 

h B. All of the heat ab- 

^ sorbed in A is given up at 

j-j B to the cooling water; 

the only wastes, in theory, 
arise from radiation. Con- 
ceive the valve c to be 
closed, and the space from 
the liquid level d to this 
valve to be filled with satu- 
rated vapor, no air being 
present in any part of the 
apparatus. Then when the 
value c is opened, a vacuum will gradually be formed throughout the 
system, and evaporation will proceed at lower and lower temperatures. 

Since the total heat of saturated vapor decreases with decrease of 
pressure, evaporation will thus be facilitated. In practice, however, the 
apparatus cannot be kept free from air ; and, notwithstanding the opera- 
tion of the condenser, the vacuum would soon be lost, the pressure increas- 
ing above that of the atmosphere. This condition is avoided by the use 
of a vacuum pump, which may be applied at e, removing air only; or, in 
usual practice, at/, removing the condensed liquid as well. Evaporation 
now proceeds continuously at low pressure and temperature. The possi- 
bility of utilizing low-temperature heat now leads to marked economy. 

592. Application. Vacuum distillation is employed on an important scale in 
sugar refineries, soda process paper-pulp mills, glue works, glucose factories, for 
the preparation of pure water, and in the manufacture of gelatine, malt extract, 



DISTILLATION 




382 



APPLIED THERMODYNAMICS 



wood extracts, caustic soda, alum, tannin, garbage products, glycerine, sugar of 
milk, pepsin, and licorice. In most cases, the multiple-effect apparatus is employed 
(Art. 594). 

593. Newhall Evaporator. This is shown in Fig. 289. Steam is used 
to supply heat ; it enters at A, and passes through the chambers A 1 , A 2 } 
to the tubes B, B. After passing through the tubes, it collects in the 
chambers C 2 , C 1 , from which it is drawn off by the trap D. The liquid 
to be distilled surrounds the tubes. The vapor forms in E, passes around 
the baffle plate F and out at G. The concentrated liquid is drawn off from 
the bottom of the machine. 

594. Multiple-effect Evaporation. Conceive a second apparatus 
to be set alongside that just described ; but instead of supplying 




Fig. 290. Art. 595. — Triple Effect Evaporator. 

steam at J., let the vapor emerging from G- of the first stage be 
piped to A in the second, and let the liquid drawn off from the bot- 



MULTIPLE-EFFECT EVAPORATION 



383 




torn of the first be led into the second ; then further evaporation may 
proceed without the expenditure of additional heat, the liquid being 
partially evaporated and the vapor partially condensed by the inter- 
change of heat in the second stage, the pressure in the shell {outside 
the tubes) being less than that in the first stage. The construction will 
be more clearly understood by reference to Fig. 290. 

595. Yaryan Apparatus. Here the heat is applied outside the tubes, 
the liquid to be distilled being inside. The liquid is forced by a pump 
through a small orifice 
at the end of the tube, 
breaking into a fine 
spray during its pas- 
sage. The fine sub- 
division and rapid 
movement of the 
liquid facilitate 
the transfer of heat. 
The baffle plates E, 
E, Fig. 291, serve to 
separate the liquid and 
its vapor, the former 

settling in the chamber b, the latter passing out at c. Figure 290 shows a 
"triple-effect" or three-stage evaporator; steam (preferably exhaust 
steam) enters the shell of the first stage. The liquor to be evaporated 
enters the tubes of this stage, becomes partly vaporized, and the separated 
vapor and liquid pass off as just described. From the outlet c, Fig. 291, 
the vapors pass through an ordinary separator, which removes any ad- 
ditional entrained liquid, discharging it back to b, and then proceed to 
the shell of the second stage. Meanwhile the liquid from the chamber b 
of the first stage has been pumped, through a hydrostatic tube which 
permits of a difference in pressure in two successive sets of tubes, into the 
tubes of the second stage. As many as six successive stages may be used ; 
the vapors from the last being drawn off by a condenser and vacuum 
pump. The liquid from the chamber b of the last stage is at maximum 
density. 

596. Condition of Operation. The vapor condensed in the various 
shells is ordinarily water, which in concentrating operations may be drawn 
off and wasted, or, if the temperature is sufficiently high, employed in the 
power plant. The condenser is in communication with the last tubes, and, 
through them, with all of the shells and tubes excepting the first shell j but 



Fig. 291. Art. 595. — Yaryan Evaporator. 



384 APPLIED THERMODYNAMICS 

between the various stages we have the heads of liquid in the chambers b, 
which permit of carrying different pressures in the different stages. A 
gradually decreasing pressure and temperature are employed, from first to 
last stage; it is this which permits of the further boiling of a liquid 
already partly evaporated in a former effect. The pressure in the tubes of 
any stage is always less than that in the surrounding shell ; the pressure 
in the shell of any stage is equal to that in the tubes of the previous stage. 

597. Theory. Let If be the weight of dry steam supplied; the 
heat which it gives up is WL. Let w be the weight of liquid enter- 
ing the first stage, H its heat, and h and I the heat of the liquid and 
latent heat corresponding to the pressure in the first-stage tubes. If 
x pounds of this liquid are evaporated in the first stage, the heat 
supplied is xl + wQi — H), theoretically equal to WL; whence 

x= [WL - wQi - #)] +1 

Then x pounds of vapor enter the shell of the second stage, giving 
up the heat xl. The weight of liquid entering the tubes of the 
second stage is w — x. Let the latent heat and heat of liquid at 
the pressure in the tubes of this stage be m and i : then the heat ab- 
sorbed, if y pounds be evaporated, is ym -f (w — x)(i — A), the last 
term being negative, since i is less than h. Then 

y = [xl —(w — x)(i — A)] -r- m. 

Consider now a third stage. The heat supplied may be taken at ym ; 

the heat utilized at 

zM+ (w - x - y)(I- 0, 

(z being the weight of liquid evaporated, M its latent heat, and I the 
corresponding heat of the liquid), 

whence z = [ym — (w — x — y)(I — z)] -r- M. 

The analysis may be extended to any number of stages. 

598. Rate of Evaporation. Ordinarily, the evaporated liquid is an aqueous 
solution ; the total evaporation per pound of steam supplied increases with the 
number of stages, being practically limited by the additional constructive expense 
and radiation loss. For a triple-effect evaporator, the total evaporation per W 
pounds of steam supplied is x + y + z. Let W = 1, and let the steam be supplied 
at atmospheric pressure, the vacuum at the condenser being 0.1 lb. absolute, and 
the successive shell pressures 14.7, 8.1, 1.5. The pressures in the tubes are then 
8.1, 1.5, and 0.1 : whence L= 970.4, / = 987.9, h = 151.3, m = 1027.8, {=81.9, 7=6.98, 



GOSS EVAPORATOR 385 

M = 1048.1. Let H be 100, the liquid being supplied at 132° F. A definite re- 
lation must exist between w and W, in order that the supply of vapor to the last 
effect, y, may be sufficient to produce evaporation, yet not so great as to burden the 
apparatus ; this is to be determined by the degree of concentration desired in any 
particular case, whence x + y + z = (/)w, in which (/) represents the proportion of 
liquid to be evaporated. Let (/) = 1.0, as is practically the case in the distillation 
of water; then w = x + y + z. We now have, x = 0.982 -0.0521 w, y = 0.88 + 0.0211 w, 
2 = 0.726 + 0.094 w, x+y+z=w= 2.588 + 0.063 w, whence w = 2.76. This is 
equivalent to about 27.6 lb. of water evaporated per pound of coal burned under 
the best conditions. By increasing the number of effects, evaporation rates up to 
37 lb. have been attained in the triple-effect machine. 

599. Efficiency. The heat expended in evaporation is in this case 
xl + ym + zM- 3080 B. t. u. The heat supplied by the steam was WL = 970.4 B. t. u. 
The efficiency is, therefore, apparently 3.18, a result exceeding unity. A large 
amount of additional heat lias, however, been furnished by the substance itself, which 
is delivered, not as a vapor, but as a liquid, at the condenser. 

600. Water Supply. The condenser being supplied per pound of steam 
supplied to the first stage with v pounds of water, its heat increasing from 
n to JV, the heat interchange is zM=v(N—n), whence, v=zM-r- (N—n), the 
liquid being discharged at the boiling point corresponding to the pressure 
in the condenser. In this case, for N— n = 25, v = 40.2 lb., or the water 
supply is 40.2 -=- 2.76 = 14.5 lb. per pound of liquid evaporated. Some ex- 
cess is allowed in practice : the greater the number of effects, the less, gen- 
erally speaking, is the quantity of water required. 

601. The Goss Evaporator. This is shown in Fig. 292. Steam enters 
the first stage F from the boiler G, say at 194 lb. pressure and 379° F. 
The liquid to be evaporated (water) here enters the last stage A, say at 
62° F. ; the boiling of the liquid in each successive stage from F to A 
produces steam which passes to the interior tube of the next succeeding 
stage, along with the water resulting from condensation in the interior tube 
of the previous stage. The condensed steam from the first stage, is, how- 
ever, returned to the boiler, which thus operates like a house-heating boiler, 
with closed circulation. Let 1 lb. of liquid be evaporated in F; its pressure 
and temperature are so adjusted that, in this case, the whole temperature 
range between that of the steam (379° F.) and that of the liquid finally dis- 
charged from A (213° F.) is equally divided between the stages. The 
amount of vapor produced in any stage may then be computed from the 
heat supplied for the assigned temperature and corresponding pressure. 
Finally, in A, no evaporation occurs, the incoming liquid being merely 
heated ; and it is found that the weights of discharged liquid and incoming 
liquid are equal, amounting each to 4.011 lb. The steam supplied by the 



386 



APPLIED THERMODYNAMICS 




FUSION 387 

boiler may be computed ; in F, we condense steam at 379° F., at which its 
latent heat per pound is 845.8. It is assumed that 3 per cent of the heat 
supplied in each effect is lost by evaporation ; the available heat in each pound 
of steam supplied is then 0.97 x 845.8 = 820.426. This heat is expended in 
evaporating 1 lb. of water at 312.6° to dry steam at 345.8°, requiring 

1187.44 - 282.26 = 905.18 B. t. u., for which 55^8 =11 lb f gteam are 

820.43 

required. The whole evaporation for the six-effect apparatus is ' = 

3.646 lb. per pound of steam. For the second effect, E, the heat supplied 
is £345.8 = 870.66, gross, or 0.97 x 870.66 = 844.54, net. The heat utilized 
is 1.873(282.22-248.7) + (0.873 x 895.18) = 844.54. In D, the heat supplied 
is 0.97[(0.873 x £ 312 . 6 ) + 1(316.98 - 282.22)] = 790.8 ; that utilized is 
2.633(248.7 -215.3) + (0.76x918.42) = 790.8. The heat interchange is 
perfect ; it should be noted that the liquid to be evaporated and the heat- 
ing medium are moving in opposite directions. This involves the use of a 
greater amount of heating surface, but leads to higher efficiency, than the 
customary arrangement. An estimated economy of 60 lb. of water per 
pound of coal is possible with seven stages (1). 

Fusion 
602. Change of Volume during Change of State. The formula 



V-v = - 



ISLdT 



T dP 



was derived in Art. 368. The specific volume of a vapor below the criti- 
cal temperature exceeds that of the liquid from which it is produced; 

d T 

consequently V— v has in all cases a positive value, and hence — must 

be positive; i.e. increase of pressure causes an increase in temperature. 
It is universally true that the boiling points of substances are increased by 
increase of pressure, and vice versa, at points below the critical tempera- 
ture. If for any vapor we know a series of corresponding values of V, L, 
T, and v, we may at once find the rate of variation of temperature with 
pressure. 

603. Fusion. The same expression holds for the change of state de- 
scribed as fusion ; the Carnot cycle, Figs. 162, 163, may represent melting 
along ab, adiabatic expansion of the liquid along be, solidification along 
cd, and adiabatic compression of the solid to its melting point along da. 
In this case, F'does not always exceed v ; it does for the majority of sub- 
stances, like wax, spermaceti, sulphur, stearine, and paraffin, which con- 
tract in freezing ; and for these, we may expect to find the melting point 



388 



APPLIED THERMODYNAMICS 



raised by the application of pressure. This has, in fact, been found to be 
the case in the experiments of Bunsen and Hopkins (2). On the other 
hand, those few substances, like ice, cast iron, and bismuth, which expand 
in freezing, should have their melting points lowered by pressure ; a result 
experimentally obtained, for ice, by Kelvin (3) and Mousson (4). The 
melting point of ice is lowered about 0.0135° F, for each atmosphere of 
pressure. The expansion of ice in freezing is of practical consequence. A 
familiar illustration is afforded by the bursting of water pipes in winter. 

604. Comments. As the result of a number of experiments with non-metallic 
substances, Person (5) found the following empirical formula to hold : 

L=(C-c)(T+25Q), 

in which L is the latent heat of fusion, C, c are the specific heats in the 
liquid and solid states respectively, and T the Fahrenheit temperature of fusion. 
Another general formula is given for metals. A body may be reduced from the 
solid to the liquid state by solution. This operation is equivalent to that of fusion, 
but may occur over a wider range of temperatures, and is accompanied by the ab- 
sorption of a different quantity of heat. The applications of the fundamental 
formulas of thermodynamics to the phenomena of solution have been shown by 
Kirchoif (6). The temperature of fusion is that highest temperature at which the 
substance can exist in the solid state, under normal pressure. The latent heat of 
fusion of ice has a phenomenally high value. 



Liquefaction of Gases 

605. Graphical Representation. In Fig. 293, let a represent the 
state of a superheated vapor. It may be reduced to saturation, and 
liquefied, either at constant pressure, along acd, 
the temperature being reduced, or at constant 
temperature along abe, the pressure being in- 
creased. After reaching the state of satura- 
tion, any diminution of volume at constant 
temperature, or any de- 
crease in temperature at 




Fig. 293. Art. 605. — Li que- constant volume, HlUSt 
faction of Superheated . . . , , . 

Vapor produce partial lique- 

faction. Constant tem- 
perature liquefaction is not applicable to gases 
having low critical temperatures. Thus, in 
Fig. 294,^6 is the liquid line and cd the Fig. 294. Art. 605.-Lique- 

° , L factiou and Critical Tem- 

saturation curve of carbon dioxide, the two peratme. 




LIQUEFACTION OF GASES 389 

meeting at the critical temperature of 88° F. From the state e 
this substance can be liquefied only by a reduction in temperature. 
With "permanent" gases, having critical temperatures as low as 
— 200° C, an extreme reduction of temperature must be effected 
before pressure can cause liquefaction. 

606. Early Experiments. Monge and Clouet, prior to 1800, had liquefied sul- 
phur dioxide, and Northmore, in 1805, produced liquid chlorine and possibly also 
sulphurous acid, in the same manner as was adopted by Faraday, about 1823, in 
liquefying chlorine, hydrogen sulphide, carbon dioxide, nitrous oxide, cyanogen, 
ammonia, and hydrochloric acid gas. The apparatus consisted simply of a closed 
tube, one end of which was heated, while the other was plunged in a freezing mix- 
ture. Pressures as high as 50 atmospheres were reached. Colladon supplemented 
this apparatus with an expansion cock, the sudden fall of pressure through the 
cock cooling the gas ; and in Cailletet's hands this apparatus led to useful results. 
Thilorier, utilizing the cooling produced by the evaporation of liquid carbon diox- 
ide, first produced that substance in the solid form. Natterer compressed oxygen 
to 4000 atmospheres, making its density greater than that of the liquid, but with- 
out liquefying it. Faraday obtained minimum temperatures of — 166° F. by the 
use of solid carbon dioxide and ether in vacuo. 

607. Liquefaction by Cooling. Andrews, in 1849, recognizing the 
limiting critical temperature, proposed to liquefy the more permanent 
gases by combining pressure and cooling. Figure 295 shows the 
principle involved. Let the gas be com- 
pressed isothermally from P to a, expanded 
through an orifice along ab, re-compressed to 
c, again expanded to d, etc. A single cycle 
might suffice with carbon dioxide, while 
many successive compressions and expansions 




would be needed with a more permanent gas. fig. 295. Art. 607.— Lique- 
The process continues, in all cases, until the ^ n by Pressure and 
temperature falls below the critical point ; 

and at x the substance begins to liquefy. The action depends upon 
the cooling resulting from unrestricted expansion. With an abso- 
lutely perfect gas, no cooling would occur ; the lines ab, cd, etc., 
would be horizontal, and this method of liquefaction could not be 
applied. The " perfect gas," in point of fact, could not be liquefied. 
All common gases have been liquefied. 

608. Modern Apparatus. Cailletet and Pictet, independently, in 1877, 
succeeded in liquefying oxygen. The Pictet apparatus is shown in 



390 



APPLIED THERMODYNAMICS 



Fig. 296. The jacket a was filled with liquid sulphur dioxide, from which 
the vapor was drawn off by a pump, and delivered to the condenser b. 

The compressor c re-delivered this 
substance in the liquid condition 
to the jacket, cooling in d a quan- 
tity of carbon dioxide which was 
itself compressed in e and used as 
a cooling jacket for the oxygen 
gas in /. The oxygen w r as formed 
in the bomb g, and expanded 
through the cock h, producing a 
fall of temperature which, sup- 
plemented by the cooling effect 
of the carbon dioxide, produced 
The series of cooling agents used suggested the name 




Fig. 296. Art. 608, Prob. 7. — Cascade System. 



liquid oxygen. 
cascade system. 



609. De war's Experiments. Dewar liquefied air in 1884 and nitrogen about 
1892. In 1895 he solidified air by free expansion, producing a jellylike substance. 
In 1896 he obtained liquid hydrogen, by the use of which air and oxygen were 
solidified, forming white masses. A temperature of — 396.4° F. was obtained. 
Dewar's final apparatus was that of Pictet, but compressors were used to deliver 
the gases to the liquefying chamber, and ethylene was employed in place of car- 
bon dioxide. 



610. Regenerative Process ; Liquid 
Air. The fall of temperature ac- 
companying a reduction in pressure 
has been utilized by Linde (7) and 
others in the manufacture of liquid 
air. In the first form of apparatus, 
shown in Fig. 297, air was com- 
pressed to about 2000 lb. pressure in 
a three-stage machine A, and after 
cooling in B was delivered to the 
inner tube of a double coil (?, through 
which it passed to the expansion 
valve D. Here a considerable fall 
of temperature took place. The 
cooled and expanded air then passed 
back through the outer tube of the 




Fig. 297. Art 



610. — Liquefaction 
Air. 



LIQUID AIR 391 

coil, cooling the air descending the inner tube, and was discharged 
at F. The effect was cumulative, and after a time liquid air was 
deposited in E. In the present type of machine, the compressor 
takes its supply from I 7 , a decided improvement. The regenerative 
principle has been adopted in the recent forms of apparatus of 
Hampson, Solvay, Dewar, and Tripler. 

The latent heat of evaporation of air at atmospheric pressure is about 140 
B. t. u. (8). In its commercial form, it contains small particles of solid carbon 
dioxide ; when these are removed by filtration, the liquid becomes clear. The 
boiling point of nitrogen is somewhat higher than that of oxygen ; fairly pure 
liquid oxygen may, therefore, be obtained by allowing liquid air to partially 
evaporate (9). The cost of production of liquid air has been carefully estimated 
in one instance to approach 22 cents per pint (10). 

(1) Trans. A. S. M. E., XXV, 03. The steam table used was Peabody's, 1890 ed. 
The temperatures noted on Fig. 292 are approximate : those in the text are correct. 
(2) Bep. B. A., 1854, II, 56. (3) Phil. Mag., 1850: III, xxxvii, 123. (4) Des- 
chanel, Natural Philosophy (Everett tr.), 1893, II, 331. (5) Ann. de Chem. et de 
Phys., November, 1819. (6) Pogg. Ann., 1858. (7) Zeuner, Technical Thermody- 
namics (Klein), II, 303-313; Trans. A. 8. M. E., XXI, 156. (8) Jacobus and Dick- 
erson : Trans. A. 8. M. E., XXI, 166. (9) See the very complete paper by Eice, 
Trans. A. 8. M. E., XXI, 156. (10) Tests of a Liquid Air Plant, Hudson and Gar- 
land ; University of Illinois Bulletin, V, 16. 

SYNOPSIS OF CHAPTER XVII 

Distillation 
The still is a device for purifying liquids or recovering solids by partial evaporation. 
By evaporation in vacuo, the heat consumed may be reduced in many important 

applications : waste heat may be employed. 
Steam may supply the heat; in the Newhall apparatus, the steam circulates through 

tubes. 
In the Yaryan apparatus, the steam surrounds the tubes. 
The vapors rising from the solution may supply the heat required in a second " effect," 

provided that the solution there is under a less pressure than in the first stage. 

As many as six stages are used, the pressure on the solution decreasing step by step. 

Evaporation per effect : x = ^L-w(h-H) = xl - (w - x) (i - h) 

I m 

__ ym — (w — x — y ) (J— i) 
Z - M~ 

In a typical case, the triple-effect machine gives an evaporation of 2.76 lb. per pound of 
steam. 

zM 

Water required at the condenser per pound of liquid evaporated = 

(N- n) 

In the Goss evaporator, the steam and the solution move in opposite directions ; this 

increases the necessary amount of surface, but also the efficiency. 



392 APPLIED THERMODYNAMICS 

Fusion 

778 L dT 
The formula V—v — — applies to fusion. The melting points of substances 

may be either raised or lowered by the application of pressure, according as the 
specific volume in the liquid state is greater or less than that in the solid state. 

The melting point of ice is lowered about 0.0135° F. per atmosphere of pressure imposed. 

L =(C— c)(T + 256) for non-metallic substances. 

Liquefaction of Gases 

A vapor below the critical temperature may be liquefied either at constant pressure or 

at constant temperature. 
No substance can be liquefied unless below the critical temperature. 
A few common substances have been liquefied by the use of pressure and freezing 

mixtures. 
A further lowering of temperature is produced by free expansion. 
Liquefaction may be accomplished with actual gases by successive compressions and 

free expansions. 
The Pictet apparatus {cascade system) employed the latent heat of vaporization of 

successive fluids to cool more volatile fluids. 
The regenerative system provides for the free expansion of a highly compressed gas 

previously reduced to atmospheric temperature. This is used in manufacturing 

liquid air. 

PROBLEMS 

1. Water entering a still at 40° F. is evaporated, (a) at atmospheric pressure, 
(6) at 2 lb. absolute pressure. What is the saving in heat in the latter case ? What 
more important saving is possible ? 

2. Water entering a double-effect evaporator at 80° F. is completely distilled, the 
steam supplied being dry and at atmospheric pressure, the pressure in the second-stage 
shell being 8 lb. and that in the second-stage. tubes 1 lb. Cooling water is available at 
60° F. The temperature of the circulating water at the condenser outlet is 80°. 
Find the steam consumption per pound of water evaporated and the cooling water 
consumption, if the vacuum pump discharge is at 85° F. 

3. In Fig. 292, take temperatures as given ; assume one pound of water to be com- 
pletely evaporated in F, and complete condensation to occur in the inner tube of each 
effect ; and compute, allowing 3 per cent for radiation, as in Art. 601 : 

(a) The weight of steam condensed in F. 

(b) The weight of steam evaporated in E, and of water delivered to E. 

(c) The weight of boiler steam used per pound of water evaporated in the whole 
apparatus. Use the steam tables on pp. 247, 248. 

4. The weight of one cubic foot of H 2 at 32° F. and atmospheric pressure being 
57.5 lb. as ice and 62.42 lb. as water, and the latent heat of fusion of ice being 142 
B. t. u., find how much the melting point of ice will be lowered if the pressure is 
doubled (Art. 603). 

5. The specific heat of ice being 0.504, find its latent heat of fusion at 32° F. from 
Art. 604. 



PROBLEMS 393 

6. How much liquid air at atmospheric pressure would be evaporated in freezing 
1 lb. of water initially at 60° F. ? 

7. In a Pictet apparatus, Fig. 296, 1 lb. of air is liquefied at atmospheric pressure, 
free expansion having previously reduced its temperature to the point of liquefaction. 
The condensation is produced by carbon dioxide, which evaporates in the jacket with- 
out change of temperature, at such a pressure that its latent heat of vaporization is 
200 B. t. u. How many pounds of carbon dioxide are evaporated ? This dioxide is 
subsequently liquefied, at a higher pressure and while dry (latent heat = 120), and 
cooled through 100° F. Its specific heat as a liquid may be taken as 0.4. The lique- 
faction and cooling of the carbon dioxide are produced by the evaporation of sulphur 
dioxide (latent heat 220 B. t. u.). What weight of sulphur dioxide will be evaporated 
per pound of air liquefied ? Why would the operation described be impracticable ? 

8. From Art. 245, find the fall of temperature at expansion in a Linde air machine 
in which the air is compressed to 2000 lb. absolute and cooled to 60° F., and then ex- 
panded to atmospheric pressure. How many complete circuits must the air make in 
order that the temperature may fall from 60° F. to — 305° F., if the same fall of tem- 
perature is attained at each circuit ? 

9. Plot on the entropy diagram the path of ice heated at constant pressure from 
— 400° F. to 32° F., assuming the specific heat to be constant, and then melted at 
atmospheric pressure. How will the diagram be changed if melting occurs at a pres- 
sure of 1000 atmospheres ? 

Plot a curve embracing states of the completely melted ice for a wide range of pres- 
sures. Construct lines analogous to the constant dryness lines of the steam entropy 
diagram and explain their significance. 

10. At what temperature will the latent heat of fusion of ice be ? What would 
be the corresponding pressure ? 



CHAPTER XVIII 

MECHANICAL REFRIGERATION 

611. History. Refrigeration by " freezing mixtures" has been practiced for 
centuries. Patents covering mechanical refrigeration date back at least to 1835 (1). 
In the first machines, ether was the working substance, and the cost of operation 
was high. Pictet introduced the use of sulphur dioxide and carbon dioxide. The 
transportation of refrigerated meats began about 1873 and developed rapidly after 
1880, most of the earlier machines using air as a working fluid. The possibility 
of safely shipping refrigerated fresh fruits, milk, butter, etc., has revolutionized 
the distribution of these food products ; and, to a large extent, refrigerating pro- 
cesses have eliminated the use of ice in breweries, packing houses, fish and meat 
markets, hotels, etc. The two important applications of artificial refrigeration at 
present are for the production of artificial ice and for cold storage. 

612. Carnot Cycle Reversed. In Fig. 298, let the cycle be 
worked in a counter-clockwise direction. Heat is absorbed along 
dc and emitted along ba; the latter quantity of heat exceeds the 
former by the work expended, abed. The object of refrigeration 
is to cool some body. This cooling may be produced by a flow of 




T 

a. ,b 

d c 

1 1 r 



Fig. 298. Art. 612. — Reversed Carnot Cycle. 



heat from the body to the working fluid along d c. Cyclic action is 
possible only under the condition that the working fluid afterward 
transfer the heat to some second body along ba. The body to be 

394 



REGENERATIVE REFRIGERATION 



395 



cooled is called the vaporizer ; the second body, which in turn re- 
ceives heat from the working fluid, is the cooler. The heat taken 
from the vaporizer is ndcN; that discharged to the cooler is nabN. 
The function of the machine is to cause heat to pass from the vapor- 
izer to a substance warmer than itself; i.e. the cooler. This is 
accomplished without contravention of the second law of thermo- 
dynamics, by reason of the expenditure of mechanical work. The 
refrigerating machine is thus a heat pump. 

The Carnot cycle, with a gas as the working fluid, would lead to an exces- 
sively bulky machine (Art. 249). Early forms of apparatus therefore embodied 
the regenerative principle (Art. 257). This 
is illustrated in Fig. 299. 

Without the regenerator, air would 
be compressed adiabatically from 1 to 
2, cooled at constant pressure along 
2 3, expanded adiabatically along 3 4, 
and allowed to take up heat from the 
body to be refrigerated along 41. In 
practice, this heat is partly taken from 
the body, and partly from other sur- 
rounding objects after the working 
air has left this body, say at 5. The 

absorption of heat along 5 1 then effects no good purpose. If, however, 
this part of the heat be absorbed from the compressed air at 3, that 
body of air may be cooled, in consequence, along 3 6, so that adiabatic ex- 
pansion will reduce the temperature to that at 7, lower than that at 4. 
This is accomplished by causing the air leaving the cooler to come into 
transmissive contact with that leaving the vaporizer. The effect of the 
regenerator is cumulative, increasing the fall of temperature at each step ; 
but since the expansion cylinder must be kept constantly colder as expan- 
sion proceeds, a limit soon arises in practice. 

In Kirk's machine (1863), a compressing cylinder was used for the operation cb, 
Fig. 298, and two expansive cylinders for the operation ad, one receiving the air 
from each end of the compressor cylinder. The pressure throughout the cycle was 
kept considerably above that of the atmosphere, and temperatures of — 39° F. were 
obtained. The regenerator consisted of layer? of wire gauze located in the pis- 
tons (2). The air machines of Hargreaves and Inglis (1878), Tuttle and Lugo, 
Lugo and McPherson, Hick Hargreaves, Stevenson, Haslam, Lightfoot, Hall, and 
Cole and Allen, have been described by Wallis-Tayler (3). The Bell-Coleman ma- 
chine may be regarded as the forerunner of all of these, although many variations 
in construction and method of working have been introduced. 




Fig. 299. Art. 612. — Regenerative 
Refrigeration. 



396 



APPLIED THERMODYNAMICS 



613. Bell-Coleman Machine. This is the Joule air engine of Art. 101, 
reversed. It operates in the net cycle given by an air compressor and an 
air engine, as in Art. 213. In Figs. 300 and 301, C is the room to be 
cooled, A a cooler, M a compressor, and N an expansive cylinder (air 
engine). In the position shown, with the pistons moving toward the left, 
air flows from C to M at the temperature T c . On the return stroke, the 
valve a closes, the air is compressed along cb, Fig. 301, and the valve s 





Fig. 300. 



Art. 613. — Bell-Coleman 
Machine. 



Fig. 301. Arts. 613, 614, 616, 622, 623. 
— Reversed Joule Cycle. 



opens, permitting of discharge into A along be, at the temperature T b . 
The operation is now repeated, the drawing in of air from O to M being 
represented by the line fc. Meanwhile an equal weight of air has been 
passing from A to A 7 " at the temperature T a , less than T h on account of the 
action of the cooler, along ea ; expanding to the pressure in C along ad, 
reaching the temperature T d , lower than that in C; and passing into C at 
constant pressure along df. The work expended in the compressor cylinder 
is fcbe; that done by the expansion cylinder is fead; the difference, abed, 
represents work required from tvithout to permit of the cyclic operation. 
If the lines ad, be, are isodiabatics, 

T h T a 



Suitable means are provided for cooling the air in the compressor cylinder, so as to 
avoid the losses due to a rise of temperature (Art. 195), and also for drying the 
air entering the expansion cylinder. 

614. Analysis of Action. Let air at 147 lb. pressure and 60° P., 
at a, Fig. 301, expand adiabatically behind a piston along ad, until 
its pressure is 14.7 lb. Its temperature at d is 



T,= T„ 



Pa 



y-\ 

= 519.6 - (10) °' 28 ' 5 = 269° absolute or 



191° F. 



Let this cold air absorb heat along dc at constant pressure, until its 



BELL-COLEMAN MACHINE 



397 



temperature rises to 0° F. Then let it be compressed adiabatically 
until its pressure is again 147 lb., along cb. Since 

% = ^, ^ = 459.6^^= 890° absolute, or 430° F. 
T d T c b \ 2b9 / 

The air now rejects heat at constant pressure along ba to cold water, 

or some other suitable agent, and the action recommences. In 

practice, the paths ad and be are not adiabatic, n < y, and the changes 

of temperature are less than those just computed. 

615. Entropy Diagram. Let aenfbc, Fig. 302, represent the pv and nt 
diagrams of a Bell-Coleman machine working in two compressive stages. 
Choosing the point c on the entropy plane arbitrarily as to entropy, but in 
its proper vertical location, we plot the line of constant pressure ca up to 
the line of temperature at a. Then ae is drawn as an adiabatic, intersected 




b 

c 


J 
n 


i 



Fig. 302. Art. 615. — Two-stage Joule Cycle. 

by the constant pressure curve ne, with nf, cb, and bf as the remaining 
paths. The area aenfbc measures the expenditure of work to effect the 
process. Along ca, theoretically, heat is taken from the cold chamber to 
the extent cgha. The work expended in single-stage compression would 
have been camb. We have then the following ratios of heat extracted to 
work expended: 

single-stage compression, c ^ ia ; two-stage compression, - 



camb 



aenfbc 



616. Work of Compression. In Fig. 301, for M pounds of air 
circulated per minute, the heat withdrawn from the cold chamber 
along dc is Q = Mk(T c — T d ). The work expended in compression is 



W c = M(p b V h + 



P h V h -P t V t . 



-PJ ? 



n 



Mn 



n 



(P b V b -P c V c ) 



Mn 



^P"V- P y\- MnP * V * 



P* 
P, 



n-\ 



398 APPLIED THERMODYNAMICS 

If compression is adiabatic, n = y, \-^) n = ttT' R c V c = RT c , 

R=k (^—^\ and W c = MkT c {^- - 1\ = Mk(T b - T c ). Similarly, 

for the engine (clearance being ignored in both cases), W E = 
Mk(T a — To). The net work expended is then 

W- W E = Mk(T b -T c -T a + T d ). 

We might also write, heat delivered to the cooler = q = Mk( T b — T a ), 

W e - W E = q- Q = Mk(T 6 - T a - T c + T d ). 

617. Cooling Water. The heat carried away at the cooler must be equal 
to the heat extracted along dc plus the heat equivalent of the net work 
expended ; it is 

Mk{T c - T d + T h - T c - T a + T d ) = Mk(T b - T a ), 

as the path indicates. Let the rise in temperature of the cooling water be 
T — t : then the weight of water required is Mk(T b — T a )-r-(T —t). 

618. Size of Cylinders. At N revolutions per impounds of air 
circulated, the displacement per stroke of the double-acting com- 

MET 

pressor piston must be, ignoring clearance, D = MV C -r- 2 JV= ^p - 

The same air must pass through the expansion cylinder ; its dis- 

MRT T 

placement is A7p d ; the two displacements have the ratio ~ if the 

cylinders run at equal r. p. m. 

The piston displacements may be corrected for clearance as in Art. 233. They 
should be further increased from 5 to 10 per cent to allow for imperfect valve action, 
etc. A slight drop in pressure at the end of expansion is not objectionable. The 
temperature T d and the capacity of the machine may be varied by changing the 
point of cut-off of the expansion cylinder. 

619. Practical Proportions. In air machines of the so-called " open type," the 
pressure in the cold chamber is that of the atmosphere ; the temperature may be 
anywhere between and 50° F. The maximum pressure is often made four at- 
mospheres absolute. The cooling water may be warmed from 60 to 80° F., and the 
air may leave the condenser at 90° F. Clearance may be from 2 per cent upward ; 
piston speeds range from 75 to 300 ft. per minute, according to the type of 
compressor. 

620. Objections to Air Machines. The size of apparatus is inordinate as com- 
pared with that of the vapor-compression machines to be described. The size may 



COEFFICIENT OF PERFORMANCE 399 

be considerably reduced by operating under pressure, as in the Kirk and Allen 
" dense air " machines, in which the suction pressure exceeds that of the atmosphere. 
Small machines of the latter type are frequently used in marine service for cooling 
pantries and for making ice for table use. The suction pressure is about 65 lb., 
the discharge pressure 225 lb. Coils must be used in the vaporizer. The regenera- 
tive modification (Art. 612) may be applied, resulting in temperatures as low as 
— 80° F. Much difficulty has been experienced in air machines from the presence 
of water vapor, w T hich congeals in the pipes and passages at low temperatures. 
Lightfoot (4) has introduced a form in which expansion is conducted in two stages. 
The temperature of the air in the first stage is reduced to only about 35° F., at 
which most of the vapor is precipitated and carried off, before the air enters the 
second cylinder. In many air machines, ordinary mechanical separators are used 
to dry the air. 

621. Coefficient of Performance. In all cases, we have the relation 
heat taken from the cold body + work done = heat rejected to the cooler ; or 
Q + W= q. The ratio Q-r- TFis described as the coefficient of performance. 
For the Carnot cycle, it is obviously t-?-(T — t), the limiting values being 
unity and infinity. This ratio is sometimes spoken of as the efficiency, a 
designation sufficiently correct so far as work expenditure goes, but which 
is apparently not in conformity with the prin- p 
ciple that no physical transformation can have 
an efficiency equalling unity. Figure 3026 
explains the anomaly. The Carnot cycle is 
abed; an and bX are indefinite adiabatics. 
Now ndcN-i- abed = Q + W may have any 
value whatever exceeding 1 ; but these two 
areas do not represent all of the heat actions 
occurring in the cycle. Heat has been re- 
moved by the condenser along ba, equivalent FlG " 302& Art 62i.-Coeffi- 

J . cient of Performance, 

to nabN= q. We may indefinitely lower the 

" efficiency " by increasing the upper temperature, as by the paths ef gh, 

etc., without at all increasing the useful refrigerating effect obtained. 

We may, in fact, regard refrigeration as a negative effect produced by the 

cooling in the condenser, the negative work done being regarded as a 

by-product of this cause : _ — q = — Q — W. A reversal of the argument 

of Art. 139 serves to show that no cycle can give a higher coefficient of 

performance than that of Carnot. 

622. Desirable Range. The value of the coefficient of performance is 
increased as that of (T—t) decreases; i.e., for efficient refrigeration, the 
range of temperature must be small, a result of extreme practical impor- 
tance. It is more economical to cool the given body of air or other sub- 
stance directly through the required range of temperature, than to cool 




400 APPLIED THERMODYNAMICS 

one tenth, say, of this body, through ten times the temperature range, 
afterward cooling the remainder by mixture. This is a special example 
of the general thermodynamic principle that mixtures of substances at 
different temperatures are wasteful, such processes being irreversible. In 
practice, T is fixed by the temperature of the cooling water. It is seldom 
less than 60° F. The refrigerant temperature t should then be kept as 
high as possible, for the service in question, if operation is to be efficient ; 
it must, however, be somewhat below the desired room or solution tem- 
perature, in order that the heat transfer may be reasonably rapid. In 
making ice, for example, t must be considerably below 32° F. 

A reversal of the demonstration in Art. 255, as applied to Fig. 801, 
shows that the coefficient of performance for the Joule cycle (Bell-Cole- 

T T 

man machine), with adiabatic paths, is — - = — ; for the corre- 
al — T d T b — T c 

sponding Carnot cycle it would have been T c -r-(T a — T c ), a naturally 

higher value.* 

Since any heat motor using air is bulky, it is necessary, in order to keep the 
size of these machines within reasonable limits, to make the temperature range 
large. This lowers the coefficient of performance, which in practice is usually 
only about one fifth that of a good ammonia refrigerating machine. Air, how- 
ever, is the least expensive of fluids, is everywhere obtainable, is safe, and may be 
worked at high temperatures without excessive pressure. 

623. The Kelvin Warming Machine. In Fig. 301, let an air engine receive its 
supply along ea at normal temperature and high pressure. The air expands along ad, 
falling in temperature, after which it is warmed by transmission from the external 
atmosphere along dc and compressed in a separate cylinder along cb. The tem- 
perature at c is equal to that at a. The compression along cb increases the tem- 
perature, and the hot air may be discharged into coils in an apartment to be 
heated. The ratio of heating done to power expended is 

n r a r a 

T h -T a -T e +T d T a -T* 

The entropy diagram is that of Fig. 302, and the ratio of heat delivered to the 
room to work expended is here bmhg -f- bmac, which exceeds unity, because of the 
heat supplied by the external air. This is consequently an ideal method for heat- 
ing. Its advantage increases as the range of temperature decreases. Considering 
an ideal heat engine and an ideal warming machine, both working in the same 
Carnot cycle, the combined efficiency so far as power is concerned would be unity. 
The efficiency would exceed that of direct stove heating without any loss whatever, 
whenever the range of temperature in the engine exceeded that in the warming 
machine. Practically, the economical range of temperature would be low, the 
machine of immense size, and the operation slow. 

* T c is the highest temperature at which refrigeration may be performed ; and T a is 
the lowest temperature at which the cooling water is effective. 



VAPOR REFRIGERATION 



401 



624. The Vapor Compression Machine. In the air machine, the temperature 
is reduced by expansion in a working cylinder. The mere flow of the air through 
a valve would not perceptibly lower its temperature (Art. 73). With a vapor, a 
decided lowering of temperature occurs when the pressure is reduced by free 
expansion. The expansion cylinder may, therefore, be omitted, and this omission 
is made in spite of the fact that an opportunity for saving some power is thereby 
lost. 

625. Principle. If a small quantity of ether be poured into the 
palm of the hand, a sensation of cold is produced. This is due to 
the rapid evaporation of the ether at the temperature of the body ; 
the heat thus absorbed by the ether is received from the hand, de- 
creasing the temperature of the latter. In Fig. 303, let the closed 




CONDENSING COIL 



Fig. 303. Art. 025. — Vapor Refrigeration. 

vessel R be partly filled with a liquid at the temperature £, having 
above it its saturated vapor. Then the pressure in R will be that 
at which the boiling point of the liquid is t. If the liquid is anhy- 
drous ammonia, for example, and t = 68° F., p == 125.056 lb. absolute. 
Let some of the liquid pass through E to the condensing coil B, in 
which the pressure is _P, less than p. Its heat per pound tends to 
change from h to ff; since h exceeds H, a certain amount of liquid 
must be evaporated in B to reestablish thermal equilibrium ; thus, 

7* = H+ XL, or X= i=i?. 



If, now, the coil B be immersed in water at a temperature higher 
than its own, the remaining (1 — X) pounds of liquid may evapo- 



402 



APPLIED THERMODYNAMICS 



rate ; the surrounding water will be cooled, giving up heat (1 — X)L 
if the substance in the coil be completely evaporated, and the pres- 
sure in B be kept constantly at P, by artificially removing the added 
vapor from B as rapidly as it is formed. The substance used must 
be one having a low boiling point even under heavy pressure, if the 
surrounding water is to be cooled much below the temperature of 
the air. 




Fig. 304. Art. 



626. — Vapor Compression 
Machine. 



626. Action of Compressor. In Fig. 304, A represents the com- 
pressor, B the condenser, the vaporizer, and D the expansion valve. 

__ The "compressor piston first 

moves upward, drawing in vapor 
from O. On the return stroke, 
the valve e is closed (the valves 
are, in practice, built in the com- 
pressor cylinder) and the vapor 
is compressed. When its pres- 
sure equals that in j9, the valve 
/ is opened, and discharge oc- 
curs. The valve /is now closed 
and D is opened, the pressure falling from that in B to that in C. 
Described as a plant cycle, vapor is compressed along cb, Fig. 305, 
condensed in the condenser along ba, becoming liquid at a, and ex- 
pands through the valve D along ad, 
its pressure falling so that it begins 
to boil violently. Further boiling 
gives the path do, along which heat 
is removed from the vaporizer C. 
Refrigeration begins at d, as soon as 
the vapor has passed the expansion 
valve. The pipes beyond this valve 
are usually covered with snow. The 
vapor process always involves (1) 
the condensation of the vapor, (2) a lowering of its pressure and 
temperature by expansion, (3) evaporation of the liquid in the 
vaporizer, and (4) compression to the initial state. The under- 
lying principles are two : the raising of the boiling point by pres- 




Fig. 305. Art. 626. — Vapor Cycle. 



VAPOR REFRIGERATION 



403 




404 



APPLIED THERMODYNAMICS 



sure, and the absorption of heat from surrounding bodies during 
evaporation. The pump analogy is useful. The vaporizer may be 
likened to a pit or well in which a fixed water level is to be main- 
tained ; by using a pump, the water may be raised to a level at which 
it will of itself flow away. The " pump " is the compressor, which 
raises the low-temperature heat of the vaporizer to a high-tempera- 
ture heat which can flow away with the condensed water. The 
heat absorbed by the water is usually valueless for further service, 
as its temperature seldom exceeds 80° F. 

Figure 306 represents a complete plant. 
The pipes a, b correspond to those similarly- 
lettered in Fig. 304. The vaporizer may- 
be merely an insulated room to be cooled, 
or a vessel of water or brine the temperature 
of which is to be lowered. There should be 
no loss of liquid in operation excepting by 
leakage. 



' / \- 


h 


/ 


/ 






k 


/ 


« 9 

! 

\m n 


J 








627. Entropy Diagram. Figure 
307 shows the various forms of en- 
tropy diagram, according as the sub- 

Fig. 307. Arts. 627, 628, 629, 630 — r J m & ' & 

Vapor Refrigeration, Entropy Dia- Stance IS wet (dcbd, dgefa) dry 

gram - (djhfoC), or superheated (djMfa) as 

it leaves the vaporizer. These are based on adiabatic paths. The 
actual operation is not a perfect Clausius cycle. During expansion 
the condition is one of constant total 
heat, giving such a path as axd, Fig. 
308. This decreases the useful re- 
frigerating effect area to ydjz. Com- 
pression may be made more economical 
than adiabatic, as in air compressors, 
by jacketing or spraying with oil or 
other liquid ; the compressive path may 
then be, say, jb, decreasing the work ex- 
penditure to axdjb, without altering the 
refrigerating effect. The path jb, if 

represented exponentially, will show a value of n less than that of 
y for the vapor in question. An actual indicator diagram from a 
vapor compressor is given in Fig. 309. 




Fig. 308. Art. 627. — Modifications 
of Refrigerative Cycle. 



VAPOR REFRIGERATION 



405 




628. Coefficient of Performance. For the cycle dcba of Fig. 307, 
in which the vapor at no state becomes superheated, maximum heat 
removed from the vaporizer is, say, 
xl. Heat is returned to it, however, 
along ad,* the liquid being lowered 
in temperature, to the extent H— h. 
The net refrigerating effect is 

Q = xl-(H-K). 

The heat delivered to the condenser is XL, and the work done is 

W=XL + R-h-xl 

The coefficient of performance is then 



Fig. 309. Art. 627. — Ammonia Com- 
pressor Indicator Diagram. 



#-<* 



H± A) _*_ (XL + H- h - xl). 



Formulas may readily be derived for the coefficient when the vapor 
becomes superheated during compression or even before compression 
begins. 



629. Multi-stage Operation : Superheat. A gain is possible by compress- 
ing in two or more stages. This gives an entropy diagram like that of Fig. 309 a. 
Fig. 307 shows that the highest coefficient of performance is attained when the 
f vapor remains saturated (wet or dry), 

throughout the cycle. Comparing the cycles 
abed and afegd, for example, the added re- 
frigeration effect cgnm is gained at the cost 
of the proportionately greater expenditure 
of external work cgefb. Superheating may 
be prevented by keeping the vapor always 
sufficiently wet at the beginning of com- 
pression, or by cooling during compression 
so as to avoid the adiabatic path, as .de- 
scribed in Art. 198. "Dry" compression 
(in which superheating occurs)- involves 
the use of jackets to permit of lubrication. 
Wet compression is far more frequently 
practiced. 




Fig. 309 a. Art. 629. — Two-stage 
Compression. 



630. Choice of Liquid. The entropy diagram, Fig. 307, shows clearly 
one consideration which should influence the choice of a working fluid. 

* In this ideal case, no cooling occurs between the condenser and the expansion 
valve or between the expansion valve and the vaporizer. 



406 



APPLIED THERMODYNAMICS 



The net refrigerating effect is reduced by the area under da, as explained 
in Art. 627. The steeper this line, the less the reduction ; the longer the 
line dc, the greater is the refrigerating effect. Steepness of the line da 
means a low specific heat of liquid; a long line dc means a high latent heat. 
The best fluids for refrigeration are therefore those in which the ratio of 
latent heat to specific heat of liquid is large. From this standpoint, ammonia 
is among the most efficient of the vapors used. With carbon dioxide, 
the area under da forms a large deduction from the gross refrigerating 
effect. 

631. Fluids Used. The vapors used for refrigeration include sulphuric ether, 
sulphur dioxide, methylic ether, ammonia, carbon dioxide, ethyl chloride, Pictet 
fluid (a mixture of carbon dioxide and sulphur dioxide), and steam. The vapor 
chosen must not be too expensive, and it must not exert a detrimental influence on 
the machinery. Ether, once commonly employed, is quite costly ; its specific volume 
is so great that the machines were excessively bulky. The inward leakage of air 
resulting from the extremely low pressures necessary often heated the compressor 
cylinder. Sulphur dioxide unites with water to form sulphurous acid, which rapidly 
corrodes the cylinder when any moisture enters the system. The Pictet fluid has been 
used only by its inventor. Carbon dioxide, though inefficient, has been commer- 
cially satisfactory excepting where its low critical temperature (Art. 379) was 
objectionable. Ammonia is the fluid principally employed ; the only serious ob- 
jection to it seems to be the presence of occasional traces of moisture. The ordi- 
nary ammonia of commerce is a weak aqueous solution of the gas, H 3 N. The 
ammonia employed in refrigerating machines is the nearly pure anhydrous lique- 
fied gas, which has an intensely irritating and dangerous odor. It boils at -26° F. 
at atmospheric pressure. 



632. Comparisons. It is interesting to compare the effects following the use 
^ of various fluids between assigned temperature limits. 

Let the cycle be one in which the vapor is dry at 
the beginning of compression, abode, Fig. 309 b. We 
have 

Q = ae f9 - gd>h = L e - (h b - h a ). 

W=q-L e . 



\ 



Fig. 309 6. Art. 632. —Dry 
Compression Cycle. 



y-i 



The value of T d is T ( 
of k is variable ; but we have 



m ' ■ 



q — gabcdf= gabh + libci + icdf 
= fo-h a + L c + k(T d - T c ). 

where y is the adiabatic exponent. The value 



k ]o Se~ = n e - n c , or k log T d - k log T c = (n e - n c ) ~ 2.3, 



COMPARISON OF VAPORS 



407 



in which T d , T c , n e 


, and n c are known. 


The following are 


specimen result 


tables, pp. 247, 248, 422, 424) : 








NH 3 


so 2 


H 2 


n 


64.4° F. 


64.4° F. 


116° F. 


T a 


5°F. 


5°F. 


32° Fo 


L c 


520.22 


153.81 


1038 


L e 


582.1 


169.745 


1060 


h 


36.86 


10.44 


84 


K 


- 25.63 


- 8.449 





P =P b 


117.42 


44.537 


1.5 


Pe = Pa 


33.667 


11.756 


0.0886 


T d 


175° F. 


159° F. 


484° F. 


k 


0.70 


0.2023 


0.493 


n e 


1.20 


0.3478 


2.1832 


n c 


1.065 


0.3140 


1.9412 


y 


1.33 


1.272 


1.298 


<i 


659.91 


191.8 


1303 


Q ' ' 


519.61 


150.86 


976 


w 


77.81 


22.05 


243 


Q + W 


6.68 


6.82 


4.02 



633. Capacity. The common basis for rating refrigerating machines 
is in tons of ice-melting effect per 24 hr. The " ice-melting " effect is a con- 
ventional term denoting the performance of 142 B. t. u. of refrigeration. 
(The latent heat of fusion of ice is approximately 142 B. t. u.) Let Q be 
the heat removed from the vaporizer per cubic foot of fluid measured at 
its maximum volume during the cycle ; then the tonnage per cubic foot is, 
theoretically, 

T=Q-j-(142x2000). 

Let D be the piston displacement, per 24 hr., in cubic feet ; then the " rat- 
ing " of the machine is 

t = DT=DQ -284000. 

In practice, this does not exactly hold, because the vapor is superheated 
by the cylinder walls during the suction stroke, its density being thus 
decreased below that of the saturated vapor. The reduction of capacity 
due to this superheating may be represented by the empirical expression 
0.04p -r- P, in which p is the pressure in the condenser, and P that at the 
vaporizer. The actual tonnage is then 



(1-0.04 I) 



DQ + 284000. 



634. Economy. A practical unit of economy is the pounds of ice-melt- 
ing effect per pound of coal burned in the boiler which drives the com- 



408 APPLIED THERMODYNAMICS 

pressor engine. The refrigerating effect per cubic foot of fluid is, if we 
ignore self -evaporation (Art. 625), 

1-0.04 jlW 

the work done in the compressor cylinder is (q — Q) ; that in the engine 
cylinder is C(q — Q), in which C is the reciprocal of the combined mechani- 
cal efficiency of engine and compressor, ranging from 1.15 to 1.25 for direct 
connected units. The foot-pounds of refrigerating effect per foot-pound 
of indicated work in the engine cylinder are then 

l-0.04j-Wc(<7-Q). 

The ice-melting effect per horse power hour is then 

1980000 A _ 0Q4 M 
142 x 778 V PJ J 

If, as in ordinary average practice, three pounds of coal are used per 
Ihp.-hr., the ice-melting effect per pound of coal is 



1980000 



1-0M^)Q+C(q-Q). 



142 x 3 x 778 V P 



635. Cooling Water. The heat absorbed by the condenser per cubic foot 
of piston displacement is 

'l_ 0.04 j^. 

The number of pounds of water required per 24 hi*, to absorb this heat, 
assuming the temperature rise of the water to be 30°, is 



A _ 0.04 ^Dg-s-30. 



The gallons of water necessary per minute for each ton of "rating" (as 
defined in Art. 633) then become, 

1.0-0.04^^ -5-/3O x 60 x 24 x 8^1 -5- j (l - 0.04 ^\QD 

-fl42x200oM. 

This is about one gallon for the given range of wa/ter temperature ; the 
usual range, however, is only about 15°. 

636. Size of Compressor. If the fluid at the beginning of compression 
be just dry, and v be the specific volume and M the weight of this dry 



COMPRESSOR DESIGN 409 

vapor circulated per minute, the total volume displaced per minute is Mv\ 
if N be the number of single strokes per minute, the piston displacement 
per single stroke of a double-acting compressor must be D = Mo -=- N. 
This must be increased for superheating, as in Art. 633, the displace- 
ment becoming 

and must be further corrected for clearance, as in Art. 233. A small 
additional increase is made in practice, to allow for the presence of air 
and moisture, etc. 

637. Compressor Design. The refrigerating effect being assigned, the nor- 
mal (unrefrigerated) vaporizer temperature and the possible condenser tempera- 
ture are ascertained. These determine the cyclic limits. The type (single- or 
double-acting) and rotative speed of the compressor are then fixed. The refriger- 
ating effect per pound of fluid under the assumed temperature conditions is now 
computed, and the necessary weight of fluid determined. The piston displace- 
ment may then be calculated and the power consumption and cooling water supply 
ascertained. 

In most vapor computations, the specific volume of the liquid may be ignored. 
This does not hold with carbon dioxide, which is worked so near its critical tem- 
perature that the specific volume of the liquid closely approaches that of the vapor. 
The losses in the vapor compressor are similar in nature, though opposite in effect, 
to those in the steam engine cylinder. The transfer of heat between cylinder walls 
and working fluid causes the most serious loss ; it is to be overcome in the same 
ways as are employed in steam engine practice. 

638. Steam Compressors. In these, the working fluid is water, injected at 
ordinary temperature into a vacuum chamber. A portion of the water vaporizes, 
absorbing heat from the remainder and thus chilling it. The vapor is then slightly 
compressed, condensed, and pumped away or back to the vaporizer. The principle 
of action is the same as that of any vapor machine, but the pressure throughout 
is less than that of the atmosphere. The temperature cannot be lowered below 
32° F. (Art. 632). 

639. Ammonia Absorption Machine. This was invented by Carre. The 
theory has been thoroughly presented by Ledoux (5); numerous develop- 
ments of the original Carre apparatus have been described by Wallis- 
Tayler (6). Instead of using the mechanical force exerted by a compressor 
to raise the temperature of the fluid emerging from the vaporizer, this 
elevation of temperature is produced by the application of external heat 
from fuel or steam coils in what is called the generator. The fluid then 
passes to the condenser, and through an expansion valve to the vaporizer. 
It cannot be returned directly to the generator, because the pressure there 
exceeds that in the vaporizer. An intermediate element, called the 



410 



APPLIED THERMODYNAMICS 



absorber, is used. The operation depends upon the well-known fact that 
water has the power of dissolving large volumes of volatile vapors ; at 
59° F., it dissolves 727 times its own volume of ammonia. This solution 
produces an exothermic reaction; heat is evolved, amounting to about 
926 B. t. u. per pound of vapor absorbed. "The mechanical force which 
draws the vapor from the vaporizer in the compression system is. here re- 
placed by the affinity of water for ammonia vapor ; and the mechanical 
force required for compressing the vapor is replaced by the heat of the 
generator, which severs this affinity and sets the vapor at liberty " (Kent). 
Ammonia is among the most soluble of the substances considered ; other 
vapors may, however, be used (7). 

640. Arrangement of Apparatus. The absorption apparatus is shown 
in outline in Fig. 310. At A is the generator, containing a strong solution 
of ammonia in water and suitably heated. The heat liberates ammonia 
gas, which passes through the pipe a to the conlsnser B. From this the 
liquefied ammonia passes out at b and is expanded through the valve h, 
taking up heat from the vaporizer C, as in the compression system. The 




Fig. 310. Art. 640. — Ammonia Absorption Apparatus. 



absorber D is a vessel containing water or a weak solution of ammonia in 
water. The solution of vapor in this water produces a suction which con- 
tinually draws vapor over from C to D. The solubility of ammonia in 
water decreases as the temperature increases, so that the evolution of heat 
in the absorber must be counteracted by jacketing that vessel with water 
or installing water coils in the solution. The waste water from the con- 
denser may be used for this cooling. The more concentrated portion of 
the liquid in D is now pumped through / to A, while the weaker solution 
is drawn off from the bottom of A and returned to the top of D through d. 
A coil heater at E provides for the interchange of heat, thus warming the 
liquid entering A and cooling that entering D, as is to be desired. 

641. Cycle. From the condenser to the vaporizer, the operation is 
identical with that in a compression plant. The absorber and generator 




ABSORPTION APPARATUS . 411 

replace the compressor. ■ Tlie rise in pressure occurs between the pump / 
and the generator outlet a. In Fig. 311, B may be taken as the state of 
the gas entering the condenser, in which it is liquefied along BA. Ex- 
pansion reduces its pressure, giving 
the path AJ. In passing through the 
vaporizer, the liquid is evaporated 
along JC. It cannot be returned di- 
rectly to the generator; nor can it 
advantageously be returned by pump- 
ing, because very little solution would 
occur at the high temperature main- 
tained in the generator. It is there- 
fore absorbed by water in D, Pig. 310, 
at a pressure nearly equal to that in Fig. 311. Arts. G41, 642.— Absorption 
0, and transferred to the generator, yc e ' 

where its pressure rises, as along CB, Pig. 311. Prom C to B, the vapor 
is in solution; but its pressure and temperature are increased by the 
application of heat, just as in compression machines they are increased at 
the expenditure of external work. The cycle is the same as that of the 
compressive apparatus. 

642. Comparison of Systems. The temperature attained at B, Fig. 311, is 

practically the same as in dry compressive system s ; it is T B = T c ( ~ J y = TA — ) 

for ammonia (y = 1.33). The refrigeration per pound of pure dry vapor is 
Q = (1 — X)L, as with the compressor. Ideally, the heat evolved in the absorber 
should be approximately sufficient to evaporate the solution in the generator. 
Actually, this heat is largely lost, on account of the necessity of cooling the ab- 
sorber. Assuming that all the steam consumed by the pumps is afterward em- 
ployed in the generator, the heat consumption of the absorption apparatus includes 
the following four items: 

R, that necessary to evaporate the cold water drained back from a portion of 
the condenser tubes ; 

E, that necessary to raise the temperature of the solution entering the genera- 
tor to that of saturation ; 

S, that necessary to distill the ammonia in the generator (latent heat plus heat 
of decomposition) ; 

W, necessary to raise the temperature of the vapor during superheating. 

Symbolically, H ■= W 4- 5 4- E + R. Items E and R may be regarded as off- 
set by the friction losses in the compressor system. We may then put H— W + S 
in the absorption system. '< A rough comparison of the two systems is as follows : 
At a suction pressure of about 34 lb. absolute, at which the vaporizer temperature 
is 5° (with ammonia), a good non-condensing steam engine will consume heat 
amounting to about 969 B. t. u. per pound of ammonia circulated, the condenser 
temperature being 65°. Under the same conditions, the absorption machine will 



412 . APPLIED THERMODYNAMICS 

consume about 72 B. t. u. in raising the temperature and about 897 B. t. u. in dis- 
tilling the ammonia ; whence H — 72 + 897 = 969. The two machines are thus 
equal in economy for a suction pressure of 34 lb." As the vaporizer temperature 
falls below 5°, the economy of the absorption system becomes better than that of 
a compressor with a non-condensing engine. The reverse is the case when the 
vaporizer temperature rises. Compared with condensing engine driven compres- 
sors, the economy is about equal for the two types when the vaporizer room tem- 
perature is zero. Where a low back-pressure is required, as in ice-making, the 
absorption system is thermodynamically superior. 

643. Steam Absorption Machines. A water-vapor machine of the class de- 
scribed in Art. 638 may dispense with the compressor, the steam being absorbed 
by and generated from solutions in sulphuric acid. This form of apparatus has 
been in use for at least a century, having been successfully developed by Carre and 
others (8). 

Details and Commercial Standards 

644. Direct Expansion. When the refrigerating fluid is itself circu- 
lated in the room or through, the material to be cooled, the system is that 
of direct expansion. While simple and economical, there are objections to 
this type of plant. The least movement of the expansion valve changes 
the lower pressure and temperature, and consequently the temperature of 
the room to be cooled. The introduction of a substance like ammonia is 
often considered too hazardous in rooms where valuable materials like 
furs would be damaged by any leakage. 

645. Brine Circulation. By expanding the refrigerating fluid in coils im- 
mersed in some harmless liquid, like salt water, the former may be kept wholly 
within the power plant ; the cooled water is then circulated through the rooms to 
be refrigerated by means of a pump. The operation is wasteful, because it in- 
volves an irreversible rise in temperature between working fluid and brine, but 
is often preferred for the reasons given. The brine serves as a "fly wheel for 
heat," smoothing out the variations in temperature which occur with direct expan- 
sion ; but a secondary circulating system is more expensive in installation and 
operation. In addition to the usual apparatus, there must be supplied a brine tank, 
which now becomes the vaporizer, coils within the brine tank, and a brine pump. 
The cooling coils in the refrigerated room, and the piping thereto, must be sup- 
plied as in direct expansion ; they are, however, rather less expensive. 

646. Fluids. Salt brine is commonly used rather than water, since the 
freezing point of the former may be as low as — 5° F. This fluid is detrimental 
to cast-iron fittings, and these are ordinarily made extra heavy when used for 
brine circulation. Chloride of calcium in solution permits of a still lower tempera- 
ture ; it may solidify at as low a temperature as — 54° F. A solution of magne- 
sium chloride is occasionally used. Salt brine cannot be left in the system after 
the circulation ceases, as the salt settles out and the freezing point is raised. 



APPLICATIONS OF REFRIGERATION 413 

647. Brine Circulation Plant. Figure 312 shows a complete plant. In opera- 
tion, the compressor is first started, drawing the air out of the pipe coils. A drum 
of anhydrous ammonia is placed at B, and the contents allowed to run into the 
liquid receiver through the valve C. The expansion valve D is then opened and 
liquid ammonia passes through to the brine tank. The valves A and F are kept 
open until the odor of ammonia is evident. They are then closed, the valve L is 
opened, and the water turned on at the condenser. The compressed vapor is now 
liquefied in the condenser, its temperature falling within 20° of that of the cool- 
ing water in usual practice. The brine pump G is started, circulating the chilled 
brine through the refrigerated room H, and the speed of the compressor is in- 
creased until the temperature of the fluid in the brine tank is about 20° below 
the required temperature in H. Ammonia is supplied at C until the level in the 
receiver remains constant. The supply is then cut off. At the beginning of the 
operation, all of the ammonia will be evaporated in E, and the vapor will be highly 
superheated during compression. As the brine is chilled, the temperature of the 
discharged vapor falls, and frost forms on the outside of the pipe /, gradually 
approaching the compressor. If the supply of fresh liquid is stopped at this point, 
superheating will continue to occur, producing " dry " compression . In " wet " 
compression, the compressor inlet becomes heavily frosted and the outlet pipe is 
sufficiently cool to be touched by the hand. With adequate jacketing, etc., the 
dry system may be in practice as economical as the wet (Art. 629), but additional 
care is necessary to avoid leakage at the stuffing boxes. A direct expansion system 
has already been shown in Fig. 306. 

648. Indirect Refrigeration. In some cases, neither brine circulating coils 
nor direct expansion coils are used in the cooling room, but air is blown over a 
bank of coils and thence through ducts to the room. This constitutes indirect 
refrigeration, providing ventilation as well as cooling. In direct refrigeration, 
provision is sometimes made for drawing off foul air by vertical flues. In certain 
applications, arrangements are made for washing or filtering and drying the air 
supply introduced. 

649. Abattoirs, Packing Houses. Refrigeration here plays an important part. 
Either direct expansion or brine circulation may be employed, the coils being 
located along the side walls near the ceiling, or suspended from the ceiling, if 
head room will permit. The latter is the better arrangement. Moisture from 
the atmosphere of the room rapidly condenses on the outside of the pipes, and 
provision must be made for removal of the drip. The atmosphere of the room 
rapidly becomes dry. 

650. Cold Storage. For preserving vegetables, fruits, poultry, eggs, butter, 
milk, cheese, fish, meats, etc., either in permanent storage or during transporta- 
tion, mechanical refrigeration has been widely applied. Temperatures of from 
25° to 40° F. are usually maintained, the temperature being lowered gradually. 
Some substances keep best when actually frozen. Mechanically cooled refrigera- 
tor cars have been described by Miller (9). For all storage-room applications, 
the fundamental principles underlying the computation of the amount and dis- 



411 



APPLIED THERMODYNAMICS 



E^^^ss^^ss^sss^ms^^sss^s^^ms^ms^s sg 



AVAVVAW 



^J 



\\\\\\\\\\^\\\\\\\\\\\\\\\^ % 



1 Q 



^ 




SO 



RbS3§ 




LtSSSSSSSSSSSSSSSSSsVS. 




\dJ 



Lv)/-///'/yy^>y^>v/> 



k\\\\\\\\\\\v\\\\\\^\\\^ 



ICE MAKING 415 

tribution of coil surface are precisely those employed in the design of heating and 
ventilating systems. Reference should be made to the works of Siebel (10) and 
Wallis-Tayler (11). The thorough insulation of the rooms and of the conduct- 
ing pipes is of much importance. 

651. Other Applications. Mechanical refrigeration is universally employed in 
breweries, for cooling of the cellars and the wort, as well as for cooling during 
fermentation (attemperator system) (12). It is popular in marine service, where 
the space occupied by stored ice, and its shrinkage, would be serious items of 
expense. It is applied in candy factories, for cooling chocolate; in candle and 
paraffin works and linseed-oil refineries for precipitating out solid waxes from 
mixtures; in dairies for cooling the milk; in tea warehouses, dynamite factories, 
in the manufacture of photographic dry plates, in wine cellars, soda-water estab- 
lishments, sugar refineries, chemical works, glue factories, and for the winter stor- 
age of furs. The losses experienced in marine transportation of cattle on the hoof 
have been greatly reduced by cooling the space between decks. Refrigeration has 
also been used for congealing quicksand during excavation and tunneling opera- 
tions in loose soil. 

A recent application is in the formation of indoor skating ponds. These are 
frozen by direct expansion through submerged coils. A fresh surface is frozen 
on whenever necessary, and this is kept smooth by the use of a planing machine. 
Pipe-line refrigeration from central stations is being practiced in at least nine 
American cities ; the present status of this public service has been studied by 
Harfc (13). 

652. Ice Making. This is one of the most important applications. The 
manufacture of ice may be carried on as an adjunct to the ordinary operation of 
any refrigerating plant. The product is from an hygienic standpoint immeasur- 
ably superior to the usual natural ice. In practice, three systems are used : the 
plate, the stationary cell, and the can, the last being of most importance. 

653. Plate System. Large, shallow, hollow, rectangular boxes are immersed in 
a tank containing the water to be frozen, dividing the body of water into narrow 
sections, corresponding to the " plates " of ice to be formed. Through the hollow 
boxes, a solution of chilled brine circulates ; in some cases, however, this brine is 
quiescent, being chilled by coils immersed in it, in which coils brine from the 
compressor plant circulates. A " plate " 14 in. thick may be produced in from 
9 to 14 days. The plates when formed are loosened by circulating warm brine for 
a few moments, and are then hoisted out by cranes. 

654. Stationary Cell System. A large number of approximately cubical 
tanks, with hollow walls and bottoms, are set in a frame. Brine is circulated 
through the w T alls. A " cake " of ice is gradually formed within the tanks. This 
is loosened in the same manner as plate ice. 

655. Clear Ice. Much difficulty has been experienced in securing a product 
free from the characteristic porous, granular structure. A clear ice has been 



416 APPLIED THERMODYNAMICS 

found to be most probable when the temperature of the operation is not too low, 
when the water is agitated during cooling, and when the layers are thin, as in the 
plate system or with shallow, stationary cells. To provide these conditions usually 
involves delay, trouble, or expense. The clear ice of the present day is pro- 
duced by the use of distilled water. This may be obtained by condensing the 
exhaust from the compressor engine, or by using that exhaust in an evaporator to 
distill in vacuo a fresh supply of water. Traces of cylinder oil must in the former 
case be thoroughly eliminated, and the water carefully filtered. 

656. Can System. The use of distilled water from the engine exhaust in 
portable cans is at present standard practice. The cans, of plain galvanized iron, 
stand in a tank containing a circulating solution of brine, the temperature being 
somewhat below 32°. Blocks of 300 lb. weight are produced in from 50 to 60 hr. 
— about one fourth the time usually required with the plate system. The ice is 
loosened by lowering the cans for a moment in warm water. The various wastes 
of water, when the condensation from the engine is employed, require that the 
amount fed the boiler shall be about 33 per cent in excess of the amount of ice to 
be made. A highly economical steam engine is thus undesirable. " The can sys- 
tem requires about one fourth the floor area and one twelfth the cubical space that 
are needed by the plate system for the same output, while it is about four times 
as rapid, and costs initially about 25 per cent less." 

In a system recently introduced, large hollow cylinders, through which ammonia 
circulates, are revolved in a freezing tank. A thin film of ice forms on the outside 
of the cylinders, and is scraped off by knives and pumped in slushy condition to a 
hydraulic press, where it is formed into cakes. The process is continuous and re- 
quires little labor. The clearness of the ice depends upon the pressure to which 
it is subjected. 

657. Details. The pressure range is usually from 190 to 15 lb. gauge approxi- 
mately. The brine may be ordinary salt brine, consisting of 3 lb. of medium 
ground salt per gallon of water (specific heat about 0.8), or calcium chloride brine, 
in the proportion of 3 to 5 lb. of chloride to one gallon, or, on the average, at about 
23° Be., weighing 13| lb. per gallon and permitting of a temperature of —9° F. 
The specific heat of this solution is about 0.9. The brine must be periodically 
examined with a salinometer. The ice-making capacity is not the same as the ice- 
melting effect described in Art. 633. To produce actual ice, the water must be 
cooled from its initial temperature to the freezing point, while the ice is usually 
formed at a temperature considerably below 32°. Roughly speaking, about one- 
half ton of actual ice may be made per ton of rated capacity. The productive 
capacity is further reduced by the losses attending the handling of the ice. 

658. Tonnage Rating. The ice-melting effect of a machine work- 
ing between the pressures^ and P is, from Art. 633, 

t = m fl- 0.04 jO^l - X)L + (142 x 2000), 
in which m is the density of the vapor at the suction pressure. 



TONNAGE RATING 417 

Since X is determined by p and P, the capacity depends directly 
upon the pressure range and the piston displacement. The Ameri- 
can Society of Mechanical Engineers (14) has standardized these 
pressures by assigning 90° and 0° F. as the corresponding tempera- 
ture limits. This makes the lowest possible room temperature about 
15° F. with direct expansion and about 25° F. with brine circulation. 
Lower temperatures are frequently required. The lower of the 
assigned temperatures also fixes the value of m. For any other 
pressures, q, Q, at the state if, x, ?, the tonnage capacity would be 

T=M(l-0Ml\l)(l--x)l+ (142x2000); whence 
Jffa-0.04lVl-a;)Z 
i m (l-0.04:Pyi-X)L 

659. Compressor Proportions. The builders of machinery do not in all 
cases rate their machines on this basis. Many of them merely state the 
piston displacement (which may range from 6500 to 8700 cubic inches per 
minute per ton of nominal capacity) or the weight of vapor circulated 
under given pressure conditions. Power rates usually range from one to 
two horse power at the engine per ton of capacity; piston speeds vary 
from 125 to 600 ft. per minute. 

660. Tests. A standard code for trials of refrigerating machines 
is under consideration by a committee of the American Society of 
Mechanical Engineers, a preliminary report having already been 
made (15). Results of tests are stated in ice melting effect in pounds 
per pound of coal or per indicated horse power hour at the compressor 
engine. Where the coal is not measured, 3 lb. of coal per hour are 
often assumed to be equivalent to one horse power. Let a be the 
ice melting effect per indicated horse power : then 

142 a - (1980000 - 778) = 0.0557 a 
is the efficiency from engine cylinder to cooling room. Let b be the 
ice-melting effect per pound of coal containing 14,000 B. t. u. ; then 

142 6-14000 = 0.01015 6 
is the efficiency from coal to cooling room. A few well-known tests 
will be quoted. 



418 APPLIED THERMODYNAMICS 

661. Air Machines. Ledoux quotes tests (16) in which the ice-melting effect 
per pound of coal was from 3.0 to 3.42 at 3 lb. coal per Ihp. ; the efficiencies from 
coal to cooling room being respectively only 0.0301 and 0.0346. A Bell-Coleman 
machine at Hamburg, tested by Schroter (17) gave from 354 to 371 calories of 
refrigeration per Ihp.-hr., the efficiency from the engine to cooling room being 
therefore from 0.551 to 0.580. The range of temperatures was very low.. About 
half the power expended in the compressor is ordinarily recovered in the expan- 
sion cylinder. 

662. Compression Machines. Most tests have been made with ammonia. 
Ledoux tabulates (18) ice-making effects per pound of coal ranging from 
9.86 to 46.29, based on 3 lb. of coal per horse power ; the corresponding 
efficiencies being from 0.10 to 0.469. A number of tests by Schroter gave 
from 19.1 to 37.4 lb., or from 0.194 to 0.379 efficiency. Shreve and Anderson 
obtained 21 lb., or 0.213 efficiency (19). Anderson and Page (20) obtained 
18.261 lb. of ice-melting effect per pound of coal containing 12,229.6 B. t. u. 
per pound ; or 65.79 lb. per Ihp. The efficiency from engine to refrigera- 
tion was 3.65; from coal, it was 0.211. The pressure range was from 
28.88 to 132.01 lb. absolute. Denton (21) reported 23.37 lb. of ice-melt- 
ing effect per pound of coal on the 3 lb. basis, working between 27.5 and 
161 lb. pressure. The ice-melting capacity for 24 hr. was 74.8 tons, the 
average steam cylinder horse power, 85 ; whence the engine to room effi- 
ciency was (23.37 x 3 x 142) -j-2545 = 3.92, and the coal to room efficiency 
about 0.236. The efficiency from coal to engine cylinder was then 
0.236 -f- 3.92 = 0.0602. A series of tests by Schroter (22) gave from 1674 
to 4444 calories of refrigeration per compressor horse power, the corre- 
sponding efficiencies being therefore from 2.61 to 6.91 ; the engine to 
room efficiency might be 15 per cent less, say from 2.21 to 5.87. A Pictet 
fluid machine (23) gave 3507 calories per horse power in the steam 
cylinder, or 5.5 efficiency. The reason for these high values, exceeding 
unity, has been stated in Art. 621. The steam engine efficiencies in none 
of these tests exceeded 15 per cent ; it did not average much over 5 per 
cent ; an average efficiency of 0.237 from coal to room corresponds to a 
coefficient of performance of about 

0.237-7-0.05=4.74 (neglecting friction of mechanism). 

The engine to room efficiency is equal to the actual coefficient of perform- 
ance multiplied by the mechanical efficiency of engine and compressor. 

663. Ammonia Absorption Machines. Assuming an evaporation of 11.1 lb. of 
water from and at 212° F. per pound of combustible, Ledoux (24) reports a test 
in which 20.1 lb. of ice-melting effect were produced per pound of coal, the over- 
all efficiency being thus 0.204. A seven-day test by Denton (25) gave 17.1 lb., 
based on 10 lb. of steam per pound of coal, the corresponding efficiency being 
about 0.173. The pressure range was from 23.4 to 150.77 lb. absolute. The tern- 



AMMONIA COMPRESSOR 



419 




420 APPLIED THERMODYNAMICS 

perature range was from 272° to 80° F. ; the coefficient of performance for the 
Carnot cycle would have been 2.83. The equivalent efficiency from coal to com- 
pressor cylinder in a compression machine must then have been at least 

0.173 -=- 2.83 = 0.0613 ; 
or from coal to engine cylinder, about 

1.2 x 0.0612 =. 0.07344. 

664. Commercial Types. Compressors may be driven directly from a steam 
cylinder, or by belt. -Any form of slow-speed engine may be used for driving ; 
a favorite arrangement is to have the steam cylinder horizontal and the ammonia 
cylinder vertical, as in Fig. 313. Tandem or cross-compound engines may be 
used. The ammonia condenser may be an ordinary surface condenser, or an 
atmospheric condenser of the form described in Art. 585, consisting of a coil of 
exposed pipes over which streams of water trickle. In other types, the ammonia 
coils are submerged in a tank of circulating water. Cooling towers are used 
where there is an inadequate water supply. 

(1) Wallis-Tayler, Refrigeration, Cold Storage, and Ice Making, 1902. (2) Zeuner, 
Technical Thermodynamics (Klein tr.), I, 384. (3) Op. cit. (4) Proc. Inst. Mech. 
Eng., 1881, 105; 1886, 201. (5) Ice-making Machines, D. Van Nostrand Co., 1906. 
(6) Op. cit., p. 154 et seq. (7) Wallis-Tayler, Op. cit., p. 25. (8) Wallis-Tayler, op. 
cit., pp. 24-32. (9) Stevens Indicator, April, 1904 ; Railroad Gazette, October 23, 
1903. (10) Compend of Mechanical Refrigeration. (11) Op. cit. (12) Op. cit., 381. 
(13) Engineering Magazine, June, 1908, p. 412. (14) Transactions, 1904. (15) Trans- 
actions, XXVIII, 8, 1249. (16) Ice-making Machines, 1906, Table A. (17) TJnter- 
suchungen an Kaltemachinen, 1887. (18) Loc. cit. (19) Wood, Thermodynamics, 
1905, 352. (20) Ibid., 348. (21) Trans. A. S. 31. E., XII. (22) Peabody, Thermo- 
dynamics, 1907, 414. (23) Schroter, Verg. Vers, an Kaltemaschinen. (24) Loc. cit. 
(25) Trans. A. S. M. E., X. 

SYNOPSIS OF CHAPTER XVIII 

A heat cycle may be reversed, the heat rejected exceeding that absorbed by the ex- 
ternal work done. 

The Ca,rnot cycle would lead to a bulky machine. Actual air machines work with a 
regenerator or in the Joule cycle. In this latter, the low-temperature heat ex- 
tracted from the body to be cooled is mechanically raised in temperature so that 
it may be carried away at a comparatively high temperature. The mechanical 
compression may occur in one or more stages. 

The Joule cycle is bounded by two constant pressure lines and two like polytropics. 
If the latter are adiabatics, 

W= Mk(T b - T c - T a + T d ), Q = 3Ik(T c - T d ), q = Mk(T b - T a ). 

The displacement per stroke of a double-acting compressor is MRT C h- 2NP c ; that of 
the engine is 3fRT d +-2XP d ; the two displacements ordinarily have the ratio 

T 

— -• These are to be modified for clearance, etc. 

T d 
Open type air machines work between pressures of 14.7 and 70 to 85 lb. ; "dense air 
machines''' 1 between 65 and 225 lb., using closed circulation and, in some cases, a 
regenerator. 



MECHANICAL REFRIGERATION 421 

Coefficient of performance = -& 5 its value usually exceeds unity ; the temperature 

T T 

range should be low. Value for Joule cvcle = ^ — = c — , if paths are adi- 

abatic. la-T d n-T c 

The Kelvin warming machine works in the Joule cycle and delivers heat proportional 

T T 

to the work expended in the ratio - — = - — , which may greatly exceed 

unity. Ta ~ Td Th ~ Tc 

The vapor compression machine uses no expansion cylinder. Refrigeration results from 

evaporation, but is reduced by the excess liquid heat carried to the cold chamber. 
The vaporizer is the body to be cooled ; the condenser removes the heat to be rejected ; 

the compressor mechanically raises the temperature without the addition of heat ; 

cooling of the fluid occurs during its passage through the expansion valve. 
The path through the expansion valve is one of constant total heat ; otherwise, the 

cycle is ideally that of Clausius. 
Q = xl — (H— h), q — XL, W = XL + II — h — xl, for vapor wet throughout com- 
pression. 
The vapor may be wet, dry, or superheated at the beginning of compression. 
The fluid used should be one having a large latent heat and small specific heat. NH 3 , 

S0 2 , and CO2 are those principally employed. 

Capacity = ice-melting effect in tons per 24 hours = ^ * 3er — -, corrected for 

F y , .. & * 142 x 2000 ' 

superheating. 

Economy = ice-melting effect per pound of coal or per Ihp.-hr. 

Calculations of economy, capacity, and dimensions must include the corrective factor 

(l-a.04|). 

The absorption machine replaces the compressor by the absorber and the generator. 
For low vaporizer temperatures it is theoretically superior to the compression 
apparatus. The absorption apparatus should give an efficiency equal to that in a 
non-condensing engine-driven compression system when the vaporizer temperature 
is 5°, and to that in a condensing engine system when it is 0°. 

Refrigeration may be indirect, by direct expansion or by brine circulation. 

In ice making the can system is more rapid and occupies less space, while costing less, 
tha^u the stationary cell or plate system. Clear ice is produced by using distilled 
water and as high a temperature as possible. An economical compressor engine 
is unnecessary. The pressure range is usually from 30 to 205 lb. The actual ice 
production is about one half the " ice-melting capacity." 

The A. S. M. E. basis for rating machines is at temperatures of 0° and 90° F. 

Usual piston displacements are from 6500 to 8700 cu. in. per minute per ton of rated 
capacity ; engine power rates, from 1 to 2 Ihp. per ton. 

Efficiency from engine cylinder to cooling room = 0.0557 x ice-melting effect per 
Ihp.-hr. 

Efficiency from coal to cooling room = 0.01015 x ice-melting effect per pound of coal 
(14,000 B. t. u.). 

Usual efficiencies from coal to cooling room, with vapor machines, range from 0.100 to 
0.469, the average in good tests being about 0.237 ; say 23| lb. of ice-melting effect 
per pound of coal. Absorption machines have not shown efficiencies quite as high ; 
those of air machines are extremely low. 



422 



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PROBLEMS 425 



PROBLEMS 

Note. Our knowledge of the properties of some of the vapors used in refrigeration 
is far from accurate. Any general conclusions drawn from the results of the problems 
are therefore to be regarded with caution. (See Art. 402.) 

1. Plot to scale to PV coordinates a Carnot cycle for air in which T= 80° F., 
t = 0° F., and the extreme range of specific volumes is from 1 to 4. Compare its area 
with that of the Joule cycle between the same volume and pressure limits. 

2. In a Bell-Coleman machine working between atmospheric and 73.5 lb. pressure, 
the temperature of the air at the condenssr outlet is 80 c , and that at the compressor 
inlet is 0°. Find the temperatures after expansion and after compression, the curves 
following the law PV 1 - 35 = c: 

3. Find the coefficient of performance for a Bell-Coleman machine with pressures 
and temperatures as given above, but with compression in two stages and intercool- 
ing to 80°. (The intermediate pressure stage to be determined as in Art. 211.) Com- 
pare with that of the single-stage apparatus. * 

4. Compare the consumption of water for cooling in jackets and condenser and 
for intercooling, in the two eases suggested. (See Art. 234.) 

5. The machine in Problem 2 is to handle 10,000 cu. ft. of free air at 32° F. per 
hour. Find the sizes of the double-acting expansion and compression cylinders ideally 
necessary at 100 r. p. m. and 400 ft. per minute piston speed. 

6. What would be the sizes of compressive cylinders, under these conditions, if 
compression were in two stages ? 

7. Find the theoretical cylinder dimensions, power consumed, coefficient of per- 
formance, and cooling water consumption, for a single-stage, double-acting, dense air 
machine at 60 r. p. m., 300 ft. per minute piston speed, the pressures being 65 and 225 
lb., the compressor inlet temperature 5°, the condenser outlet temperature of air 95°, 
and the circulating water rising from 65° to 80°. The apparatus is to make ^ ton of 
ice per hour from water at 65°. The curves follow the law pv 1 -* 5 = c. 

8. Find the theoretical coefficient of performance of a sulphur dioxide machine 
working between temperatures of 64.4° and 5° F., the condition at the beginning of 
compression being, (a) dry, (b) 60 per cent dry. Also (c) if the substance is dry at 
the end of compression. 

9. Check all values in Art. 632. 

10. What is the theoretical ice-melting capacity of the machine in Problem 5 ? 

11. Find the ice-melting capacity per horse power hour in Problem 7. 

12. Find the results in Problem 7 for an ammonia machine working between 5° and 
95°, the vapor being just dry at the end of adiabatic compression. How do the coeffi- 
cients of performance in the two cases compare with those of the corresponding Carnot 
cycles ? 

13. What is the loss in Problem 2, if a brine circulation system is employed, re- 
quiring that the temperature at the compressor inlet be — 25° F. ? 

* The formula for coefficient of performance of the Joule cycle, given in Art. 622, 
will be found not to apply when the paths are not adiabatic. 



426 APPLIED THERMODYNAMICS 

14. In a Kelvin warming machine, the temperature limits for the engine are 
300° F. and 110° F. ; those for the warming cycle are 150° F. and 60° F. Assume that 
the cycles are those of Carnot, and introduce reasonable efficiency ratios, determining 
the probable efficiency (referred to power only) of the entire apparatus. 

15. Compare the coefficient of performance in Problem 12 with those in which the 
vapor is (a) 80 per cent dry, (6) dry, as it leaves the vaporizer. 

16. Find the coefficient of performance in Problem 15 (&) if the compressive path 
is PV 1 - 25 = c. (Compare the Pambour cycle, Art. 413.) 

17. Compare the ratio latent heat afc &Q and MA o F ^ Art 632 n for 

specific heat of liquid 
ammonia, carbon dioxide, and sulphur dioxide. Draw inferences. 

18. Plot on the entropy diagram in Problem 12 the path of the substance through 
the expansion valve, determining five points. 

19. Find the temperature at the generator discharge of an ammonia absorption 
machine, the liquid from the absorber being delivered at 110° F. and 30 lb. pressure, 
and the pressure of vapor leaving the generator being 198 lb. 

20. An ammonia compression apparatus is required to make 200 tons of ice per 
24 hr.; in addition it must cool 1,000,000 cu. ft. of air from 90° to 40° each hour by 
indirect refrigeration. Making allowances for practical imperfections, find the tonnage 
rating, cylinder dimensions, power consumed, cooling water consumption, and ice- 
melting effect per Ihp.-hr., the machine being double-acting, 70 r. p. m., 560 ft. per 
minute piston speed, operating between 33.67 and 198 lb. pressure with vapor dry at 
the end of adiabatic compression, water being available at 65°. Estimate whether the 
exhaust steam from the engine will provide sufficient water for ice making. 

21. Make an estimate of the production of ice per pound of coal in a good plant. 

22. What is the tonnage rating of the machine in Problem 20 on the A.S.M.E. 
basis ? 

23. Coal containing 13,500 B. t. u. per pound drives a simple non-condensing 
engine operating an ammonia compression apparatus. The ice-melting effect is 81 lb. 
per Ihp.-hr. at the engine cylinder. The coal consumption is 3 lb. per Ihp.-hr., and the 
mechanical efficiency of the combined engine and compressor is 0.80. Find the ice- 
melting effect per pound of coal, the coefficient of performance, the efficiency from fuel 
to engine cylinder, and the efficiency from fuel to refrigeration. . May this last exceed 
unity ? 

24. An absorption apparatus gives an ice-melting effect of 1.8 lb. per pound of 
dry steam at 27 lb. pressure from feed water at 55° F. Prove that this performance 
may be excelled by a compression plant. 

25. Find a relation between coefficient of performance and ice-melting effect per 
Ihp.-hr. at the compressor cylinder. 

26. Find the tonnage on the A.S.M.E. standard basis of a 12 x 30 inch double- 
acting compressor at 60 r. p. m., using (a) ammonia, (6) carbon dioxide. 

27. Find the A.S.M.E. tonnage rating for an ammonia absorption apparatus work- 
ing between 30 and 182.83 lb. pressure with 10,000 lb. of dry vapor entering the gen- 
erator per hour. 

28. Check all derived values in Art. 660 to Art. 663. 



PROBLEMS 



427 



29. Compare the coefficients of performance, in Art. 632 and in Problem 8, with 
those of the corresponding Carnot cycles. 

30. Compute the value of X in Art. 658. 

31. Compute as in Art. 632 the results for carbon dioxide. Why might not ether 
be included in a similar comparison ? 

32. Ether at 52° F. is compressed adiabatically to 232° F., becoming wholly liquid. 
What was its initial condition ? (Fig. 315.) 

PRESSURE^ LBS. PER.SQ. IN. 




0.20 0.25 

ENTROPY 



Fig. 315. —Entropy Chart for Ether. 



33. Discuss variations with temperature of the total heat of ammonia, sulphur 
dioxide, carbon dioxide. (See tables, pp. 422-424.) 

34. Plot a total-heat entropy diagram for carbon dioxide. 

35. Find the ratio cubic inches of P iston displacement per minute for the A s M E> 

rated tonnage 
temperature limits, with vapor dry at the beginning of compression. (Art. 659.) 

36. Find a general expression for the coefficient of performance in the Joule cycle, 
the paths not being adiabatic. 

37. Discuss the economy and general desirability of using the exhaust steam from 
the engine driving an ammonia compressor, to distill in vacuo the water from which ice 
is to be made. 



428 



APPLIED THERMODYNAMICS 









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H3HN3HHVJ 3aniva3d^3i 



AMMONIA ENTROPY DIAGRAM 



429 



20 



60 



Pressure, Lbs. per Sq. In. 
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200 220 240 




Entropy above 32 °F., B.t.u: 
Fig. 316. — Entropy Chart for Ammonia. 



INDEX 



(Eeferring to Art. Nos.) 



Absolute temperature, 152-156, 167. 
Absolute zero, 44, 45, 156. 
Accumulator, 541. 

Adiabatic, 83, 100-105, 168, 173, 176, 325. 
Adiabatic expansion of steam, 372, 373. 416, 

431, 432, 513, 515, 517-520. 
Adiabatics of vapors, 391-397. 
Afterburning, 325. 
Aftercooler, 208. 
Air, Carnot cycle for, 249. 
liquid, 246, 609-610. 
specific heat, 71, 72. 
Air compressor, 193-212, 215, 216, 221- 

242, 540. 
Air cooling in compressor, 199. 
Air engine, 177, 180-192, 245, 254. 
Air refrigerating machine, 227, 612-620, 661. 
Air supply, boiler furnace, 560, 563-567, 

573, 575. 
gas engine, 309. 
Air thermometer, 41 , 42, 48, 49, 152. 
Air transmission, 243, 245. 
Alcohol engine, 279, 280, 341. 
Alcohol thermometer, 7. 
Ammonia, 403, 606, 630-632, 644, Table, 

p. 422, Fig. 316. 
Ammonia absorption machine, 639-643, 

663. 
Analysis of producer gas, 285. 
Andrews' critical temperature, 379, 380, 605, 

607. 
Anthracite coal, 560. 
Apparent ratio of expansion, 450. 
Apparent specific heat, 61. 
Atkinson gas engine, 276, 296, 297. 
Atmospheric condenser, 664. 
Atomic heat, 59. 
Automatic engine, 507. 
Automobile engine, 335, 340, 348. 
Auxiliaries, gas producer, 281. 
Avogadro's principle, 40, 53, 56. 

Back pressure, 448, 459. 

Barometric condenser, 584. 

Barrel calorimeter, 489. 

Bell-Coleman refrigerating machine, 613- 

620, 661. 
Bicycle, motor, 340. 
Binary vapor engine, 483. 
Blast furnace gas, 276, 278, 329, 353. 
Blowing engine, 179. 



Boiler, 566, 568-574. 

Boiler efficiency, 569, 571-574. 

Boiler horse-power, 570. 

Boiler surface efficiency, 574. 

Boulvin's method, 455, 456. 

Boyle's law, 38, 39, 84. 

Brake horse power, 555. 

Brauer's method, 117. 

Brayton cycle, 299, 300, 302, 304. 

Brine, 646, 657. 

Brine circulation, 645-647. 

British thermal unit, 22. 

Bucket, 512, 527-530. 

Caloric theory, 2, 131. 

Calorie, 23. 

Calorimeter, 488-494. 

Calorimetric testing of steam engine, 504, 

505, 511. 
Capacity, air compressor, 222-229, 230- 
237. 

air engine, 182, 183, 188, 190. 

air refrigerating machine, 616, 618, 619. 

compound steam engine, 476-477. 

gas engine, 277, 330. 

hot-air engine, 248, 249, 275, 277. 

Otto-cycle gas engine, 293. 

steam cycles, 418. 

steam engine, 446, 447, 449. 

vapor compressor, 633, 636, 637. 
Carbon dioxide, 379, 402, 605-608, 611, 

630-632, 637, Fig. 314, Table, p. 423. 
Carbureted air, 279. 
Carburetor, 279, 282, 310, 336. 
Cardinal property, 10, 76, 81, 88, 160, 162, 

169, 176, 370, 399. 
Carnot, 28, 130. 
Carnot cycle, 128-143, 451. 

air engine, 250. 

entropy diagram, 159, 166. 

for air, 249. 

for steam, 406. 

reversed, 138, 612, 621. 
Carnot function, 155. 
Cascade system, 608. 
Centigrade heat unit, 23. 
Centigrade thermometer, 8. 
Change of state, 15-18. 
Characteristic equation, 10, 50, 84, 363, 

390, 401, 403, 404. 
Characteristic surface, 84. 



431 



432 



INDEX 



Charles' law, 41-49, 84. 

Chimney, 575. 

Circulation in steam boiler, 569. 

Clapeyron's equation, 368. 

Clausius cycle, 408, 410, 447, 514, 544. 

Clearance, 188. 

air compressor, 222, 223. 

gas engine, 313, 324. 

steam engine, 450, 451, 462. 

vapor compressor, 616, 618. 
Clerk's gas engine, 300, 303-305. 
Closed feed-water heater, 581. 
Closed hot-air engine, 248, 275. 
Coal, 560, 578. 
Coal gas, 276, 278, 329. 
Coefficient of performance, 621, 622, 628. 
Coil calorimeter, 490. 
Combined diagrams, 466, 469-473, 475. 
Combustion, 560, 563-567, 569, 573, 575. 
Complete pressure gas engine cycle, 300, 

303-305. 
Compound steam engine, 438, 459-483, 510, 

550. 
Compound locomotive, 510. 
Compressed air, 177-247. 

distributing system, 212-221. 

refrigeration by, 227. 

storage system, 185, 245. 

transmission, 243-245. 

uses, 177, 178. 
Compression, in air compressor, 195-211. 

air engine, 189, 191. 

Carnot cycle, 132, 134. 

gas engine, 276, 295, 297, 299, 312, 313, 
325, 348. 

steam engine, 451, 462. 
Compressive efficiency, 213. 
Compressor, air, 193-212, 215, 216, 221- 
242, 540. 

vapor, 624-638, 642, 658, 660, 662, 664. 
Condensation in steam cylinder, 428-443, 

448, 460. 
Condenser, 502, 584, 585, 617, 635, 664. 
Constant dryness curve, 369. 
Constant heat curve, 370, 398. 
Constant volume curve, 377. 
Constant weight curve, 365. 
Constrained expansion, 124. 
Cooling of gas engine cylinder, 312, 314- 

318, 325. 
Cooling tower, 585, 664. 
Cooling water in refrigerating plant, 617, 

635. 
Coordinate diagrams. 81-127, 158. 
Criterion of reversibility, 139-141, 144-149. 
Critical temperature, 379, 380, 605, 607. 
Cross-compound steam engine, 464, 470, 

472. 
Curtis steam turbine, 524, 531, 537. 
Cushion air, 262, 264. 
Cushion steam, 453, 457, 575. 
Cycle, Carnot, 128-143, 451, 



Cycle, external work, 89. 

forms, 130. 

heat engine, 129. 
Cycle, heat expended in, 90. 

regenerative, 259. 

reversed, 90. 

reversible, 138-141, 147, 148, 152, 175, 
176. 
Cycles, air : 

air compressor, 194-211. 

air engine, 180-183. 

air refrigeration, 615. 

air system, 218-221. 

Bell-Coleman, 615. 

Ericsson, 270. 

hot-air engine, 256. 

Joule, 254, 255, 613, 622. 

Lorenz, 252. 

poly tropic, 251. 

regenerative air engine, 259. 

Reitlinger, 253. 

Stirling, 264. 
Cycles, gas, 276, 287-308. 

Brayton, 299. 

Clerk, 300, 303-305. 

complete pressure, 300, 303-305. 

Diesel, 306, 307. 

Lenoir, 298, 300, 301, 304. 

Otto, 276, 287-297, 300, 309-329. 

two-stroke, 289-292, 329. 
Cycles, refrigerative : 

air machine, 254, 255, 613, 615, 622. 

regenerative, 259, 610, 612. 

vapor machine, 627. 
Cycles, steam, 417, 418, 422-458. 

binary, 483. 

Clausius, 408-410, 417, 447, 514, 544. 

non-expansive, 412, 417, 423. 

Pambour, 413, 417. 

Rankine, 411, 417, 424, 429, 447. 

superheated, 414-418. 

turbine, 514. 
Cylinder condensation, 428-443, 448, 460. 
Cylinder efficiency, 212, 215, 216, 229. 
Cylinder feed, 453, 475. 
Cylinder ratios, 476, 477, 480. 
Cylinder walls, 429, 431, 432, 504, 505. 

vapor compressor, 637. 

Dalton's law, 40. 

Davis' method for determining H, 360, 388. 

De Laval steam turbine, 512, 524, 530, 

536. 
Dense air refrigerating machine, 620. 
Desormes' apparatus, 110. 
Diagram, coordinate, 81-127, 158. 

entropy, 158-160, 164, 166, 169-171, 
174, 184, 218-221, 255, 266, 307, 347, 
367, 376-378, 398, 453-458, 475, 615, 
627. 
indicator, 437, 452, 454, 486-487, 500, 
501. 



INDEX 



433 



Diagram, indicator, gas engine, 311. 

Mollier, 399, 516, 532. 

of energy, 86-90. 

temperature-entropy, 158-160, 164, 166, 
169-171, 174, 184, 218-221, 255, 266, 
307, 347, 367, 376-378, 398, 453-458, 
475, 615, 627. 

total heat-entropy, 399, 516, 532. 

total heat-pressure, 399. 

velocity, 527-529, 534. 
Diagram factor, 329, 446, 475, 633. 
Diesel engine, 306, 307. 
Difference of specific heats, 65, 67, 77, 165. 
Direct expansion, 644. 
Disgregation work, 3, 12, 15-17, 53, 56, 

64, 75, 76, 78, 80, 359, 360. 
Dissipation of energy, 176. 
Dissociation, 63, 318, 325. 
Distillation, 591-601. 
Distribution of work, compound steam 

cylinders, 464, 467, 468, 478. 
Double-acting engine, 423. 
Draft, 560, 567, 575-577, 582. 
Drop, 181, 436, 447, 465, 467, 468, 479. 
Dry compression, 629, 647. 
Dry vacuum pump, 237, 584. 
Dryness curve, 369. 
Duplex compressor, 239. 
Duty, 503. 

Economizer, 282, 582. 
Effects of heat, 12-17. 
Efficiency, air engine, 180, 185, 190, 192. 

boiler, 569, 571-574. 

boiler furnace, 574. 

Brayton cycle, 299. 

Carnot cycle, 135, 136, 142, 166. 

Clausius cycle, 409. 

compressed air system, 212-217. 

compressive, 213. 

Diesel engine, 307. 

Ericsson engine, 248, 249, 269-273. 

gas engine, 334, 342-346. 

gas producer, 284-286. 

heat engine, 128, 142, 143, 149. 

injector, 588, 590. 

Joule air engine, 235. 

Lenoir cycle, 298, 301. 

mechanical, 212, 214, 216, 342, 345, 487, 
503, 511, 546, 554-559. 

multiple-effect evaporation, 599. 

non-expansive cycle, 412. 

Otto gas engine, 295-297, 300. 

Pambour cycle, 413, 417. 

plant, 503. 

Rankine cycle, 411. 

refrigerating machine, 621, 622, 634, 
642, 661-663. 

refrigerating plant, 621, 622, 628. 

steam engine and turbine, 546, 553. 

steam turbine, 526, 529. 

Stirling engine, 265, 267, 268. 



Efficiency, superheated cycles, 415. 

thermal, 342. 

transmissive, 212-216, 243, 244. 
Efficiency, volumetric, 222-229. 
Ejector, 587. 
Electrical ignition, 323. 
Electrical resistance pyrometer, 9. 
Electric calorimeter, 494. 
Energy, 10, 12, 76-78, 81, 100, 109, 113, 

119-123, 359, 374, 375. 
Engine, air, 177, 180-192, 245. 

binary vapor, 483. 

blowing, 179. 

Clerk's, 300, 303-305. 

Diesel, 306-307. 
* gas, 276, 277, 287-308, 312, 313, 324, 
325, 330, 348. 

heat, 128, 130, 132, 139, 142, 143. 

hot-air, 248-275, 277. 

internal combustion, 248, 276, 277, 287- 
308. 

Joule, 235, 254. 

oil, 276, 279, 280, 299. 

rotary steam, 177, 192. 

steam, 408-419, 422-511, 514, 544, 550. 

turbine, 239, 512-542, 552, 553, 556. 
Entropy, 10, 157-176. 

formulas, 169. 

gases, 169. 

physical significance, 160. 

units, 171. 
Entropy diagram, 158, 174. 

air engine, 184. 

Bell-Coleman machine, 615. 

Carnot cycle, 159, 166. 

compressed air system, 218-221. 

Diesel engine, 307. 

gas engine, 347. 

Joule engine, 255. 

specific heats of gases, 164. 

steam, 398. 

steam engine, 453-458, 475. 

steam formation, 367, 376-378. 

Stirling engine, 266. 

vapor refrigeration, 627. 
Equation, characteristic, 10, 50, 84, 363, 
390, 401, 403, 404. 

of condition, 10, 50, 84, 363, 390, 401, 
403, 404. 
Equation of flow, 522. 
Equivalent evaporation, 361, 367, 386, 

572. 
Ericsson hot-air engine, 270. 
Ether, 371, 372, 402, 483, 611, 631, 663, 

Fig. 315. 
Evaporation, factor of, 361, 367, 389, 572. 

in vacuo, 591-601. 

latent heat of, 359, 360. 

rate of, 569. 
Evaporative condenser, 585. 
Evaporator, 593, 595, 600, 601. 
Exhaust line, gas engine diagram, 326. 



434 



INDEX 



Exhaust steam injector, 589. 
Exhaust steam turbine, 541. 
Expansion, constrained, 124. 

direct, 644. 
Expansion, free, 73, 75, 79, 124-127, 513, 
515, 517, 607, 610. 

latent heat of, 58, 107. 

regenerative, 610, 612. 

steam cylinder, 428-447, 450, 473, 486, 
558. 

steam turbine, 513, 515, 517. 
Expansion curve, gas engine, 325. 
Explosion waves, 319, 325. 
Exponential equation, 391, 394, 395. 
External work, 14, 15, 86-90, 95, 98, 121- 

123, 160, 359, 374, 375. 
Externally fired boiler, 568. 

Factor, heat, 170. 

Factor of evaporation, 361, 367, 389, 572. 

Feed pump, 586. 

Feed-water heater, 580-582. 

Figure of merit, 286. 

Fire-tube boiler, 568, 569. 

First law of thermodynamics, 28-37, 79, 

128, 167, 505. 
Fixed point, 6, 16, 18. 

Flame propagation, 309, 310, 319, 320, 325. 
Flow, equation of, 522. 

in nozzle, 521-523. 

in orifice, 523. 
Fluid friction, 326, 342. 
Forced draft, 577. 

Formation of steam, 354-360, 366, 381, 386. 
Free expansion, 73, 75, 79, 124-127, 513, 

515, 517, 607, 610. 
Freezing mixtures, 15, 611. 
Friction, fluid, 326, 342. 

in Joule's experiment, 76, 127. 

in nozzles, 518-520, 523. 

in steam engine, 555-559. 

in turbine buckets, 527. 
Fuel oil, 280. 
Fuels, 560, 561. 
Function, Carnot's, 155. 

thermodynamic, 170. 
Furnace efficiency, 574. 
Fusion, 602-604. 

Gas, coal, 276, 278, 329. 

liquefaction of, 605-610. 

natural, 276, 278, 329. 

oil, 279. 

perfect, 39, 50, 51, 53, 56, 74, 80, 607. 

permanent, 16, 63, 605. 

producer, 276-286, 312, 329. 

steam, 357, 390, 391. 

water, 278, 281, 329. 
Gas engine, 276, 277, 287-308, 312, 313, 
324, 325, 330, 348. 

Clerk's, 300, 303-305. 
Gas engine design, 330-335. 



Gasoline, 279, 280. 

Gas power, 276-353. 

Gas producer, 276-286. 

Gas producer auxiliaries, 281. 

Gas transmission, 276. 

Gas turbine, 540. 

Gases, kinetic theory, 53-56, 80. 

Gay-Lussac's law, 41-49. 

Goss evaporator, 601. 

Governing, air compressor, 238. 

gas engine, 336-338, 348, 349. 

steam engine, 462, 478. 
Gram-calorie, 23. 
Gravity return drip system, 583. 

H, 359, 360, 388. 

Heat absorbed, graphical representation, 

106, 123, 167. 
Heat, mechanical theory, 2-5. 
Heat balance, 346, 496. 
Heat drop, 515-519. 

Heat engine, 128, 130, 132, 139, 142, 143. 
Heater, feed-water, 580-582. 
Heat factor, 170. 
Heat of liquid, 359. 
Heat unit, 20-23. 
Heat weight, 170, 172. 
Heating surface, 569, 574. 
High-speed steam engine, 434, 507. 
High steam pressure, 143, 444, 459, 462. 
Hirn, 32. 

Hirn's analysis, 504, 505, 511. 
Hit-or-miss governing, 349. 
Horse power, boiler, 570. 

brake, 555. 
Hot-air engine, 248-275, 277. 
Hot-air jacket, 439. 
Hot-tube ignition, 322, 336, 337. 
Hydraulic compressor, 241. 
Hydraulic piston compressor, 240. 
Hydrogen, 60, 609. 

in producer gas, 284, 285, 312. 
Hyperbolic curve, 445, 450, 473, 486. 

Ice, 2, 85, 602-604. 
Ice making, 652-657. 
Ice-melting effect, 634. 
Ignition, 314-323, 325, 336, 337. 
Impulse turbine, 524, 530-533, 536-538. 
Incomplete expansion, 181, 436, 447, 465, 

467, 468, 479. 
Indicated thermal efficiency, 342. 
Indicator, 424, 484-485. 
Indicator diagram, 437, 452, 454, 486-487, 

500-501. 
gas engine, 311. 
Indirect refrigeration, 648. 
Induced draft, 577. 
Initial condensation, 430, 433, 436, 437, 

442, 448, 460. 
formula for, 437. 
Injector, 587-590. 



INDEX 



435 



Injector condenser, 584. 

Injection of water, 195, 200. 

Intercooler, 206, 207. 

Intermediate compound, 480. 

Internal combustion engine, 248, 276, 277, 

287-308. 
Internal energy, 10, 12, 76-78, 81, 100, 

109, 113, 119-123, 359, 374, 375. 
Internal work of vaporization, 359, 360. 
Internally fired boiler, 568, 569. 
Inversion, 373, 395, 401. 
Irreversibility, 11, 73-76, 78, 175, 176. 
Irreversible process, 124-127, 160, 426, 513. 
Isentropic, 168, 176. 
Isodiabatic, 108, 112. 
Isodynamic, 83, 96, 120-122. 
Isodynamic, vapor, 382. 
Isoenergic, 83, 96, 120-122, 382. 
Isometric, 83. 
Isopiestic, 83. 
Isothermal, 78, 83, 91-95, 122, 366. 

Jacket, gas engine, 352, 353. 

hot-air, 439. 

steam, 413, 438-441, 475, 482, 505. 

vapor compressor, 635. 
Jet condenser, 502, 584. 
Joule air engine, 254. 
Joule apparatus, 2, 30. 
Joule cycle, 254, 255, 613, 622. 
Joule experiment, 73-80, 124-127, 156, 176. 
Joule's law, 75-80, 109. 

Kelvin scale of absolute temperature, 153- 

156, 167. 
Kelvin warming machine, 623. 
Kerosene, 279, 280. 
Kinetic theory of gases, 53-56, 80. 
Kirk air refrigerating machine, 612. 
Knoblauch and Jakob, 384. 

Lagging, 439. 

Latent heat, of expansion, 58, 107. 

of fusion, 602-604. 

of evaporation, 359, 360. 
Lenoir cycle, 298, 300, 301, 304. 
Linde apparatus, 246, 610. 
Line of inversion, 373. 
Liquefaction of gases, 605-610. 

of steam during expansion, 372, 373, 431, 
432. 
Liquid air, 246, 609, 610. 
Locomotive boiler, 568. 

superheater, 443, 509, 554. 

tests, 497, 511, 554. 

theory, 509. 

turbo-, 540. 

types, 509, 510. 
Loop, steam, 583. 
Lorenz cycle, 252. 
Losses in steam boiler, 566. 

in steam turbine, 514. 



Mariotte's law, 38, 39. 

Marine boiler, 568. 

Marine turbine, 540. 

Mathematical thermodynamic method, 400, 

401. 
Mayer, 29, 72. 
Mayer's principle, 94. 

Mean effective pressure, 331, 446, 476, 486. 
Mean specific heat, 61, 164. 
Mechanical draft, 576, 582. 
Mechanical efficiency, 212, 214, 216, 487, 
503, 511, 546, 554-559. 

gas engine, 342, 345. 
Mechanical equivalent of heat, 2, 28-37, 

79, 505. 
Mechanical theory of heat, 2-5. 
Mercurial thermometer, 7. 
Metallic pyrometer, 9. 
Mixtures, 20, 21, 25. 

freezing, 15, 611. 

in gas engine, 309, 310, 348. 
Molecular heat, 59. 
Mollier diagram, 399, 516, 532. 
Mond gas, 278, 283. 
Motor-bicycle, 340. 
Multiple-effect evaporation, 594-601. 
Multiple expansion, 438, 459-483, 550. 
Multi-stage air compression, 205-211, 221, 

226, 232, 234, 235, 239. 
Multi-stage vapor compression, 629. 

n, 91, 97, 115-118, 164. 
Natural gas, 276, 278, 329. 
Negative specific heat, 115, 371. 
Negative work, 87, 89, 99. 
Neutrals, 319, 320. 
Newhall evaporator, 593. 
Non-expansive cycle, 412, 423. 
Nozzle, 512-515, 518-523, 525. 

Oil engine, 276, 279, 280, 299, 306, 307. 

Ofi fuel, 280. 

Ofi gas, 279. 

Open feed-water heater, 581. 

Opposed beam engine, 464. 

Optical pyrometer, 9. 

Orifice, 523. 

Otto cycle, 276, 287-297, 300, 309-329. 

Overload capacity, gas engine, 330, 333. 

steam engine, 447. 
Oxygen, 606, 608, 609. 

Pambour cycle, 413. 

Parsons turbine, 524, 533, 539, 556. 

Path, 83, 85, 88, 97-99, 111-118. 

Paths of vapors, 392-399. 

Pelton bucket, 529. 

Perfect gas, 39, 50, 51, 53, 56, 74, 80, 607. 

Permanent gas, 16, 63, 605. 

Pictet apparatus, 608. 

Pictet fluid, 631. 

Piston speeds, gas engine, 320. 



436 



INDEX 



Plant efficiency, 503. 
Pneumatic tools, 178. 
Poly tropic cycle, 251. 
Polytropic paths, 97-99, 111-118, 125, 161, 

164, 165. 
Porous plug experiment, 73-80, 124-127, 

156, 176. 
Power plant, steam, 407, 408, 560-590. 
Preheater, 186, 187. 

Pressure, high steam, 143, 444, 459, 462. 
Pressure-temperature relation, 355, 358, 

362, 368. 
Pressure turbine, 524, 533-535, 539. 
Problems, pages 10, 17-18, 28, 38, 60-62, 69, 

75-76, 88-89, 127-128, 144-145, 195- 

198, 252-255, 312-316, 347-349, 358- 

359, 378-379, 392-393, 425-427. 
Producer, 276-286. 
Producer gas, 276-286, 312, 329. 
Propagation of flame, 309, 310, 319, 320, 

325. 
Properties of steam, 360, 367, 376, 405, 420, 

421. 
Pulsometer, 506. 
Pump, feed, 586. 
pulsometer, 506. 
turbo-, 540. 

vacuum, 236, 237, 584, 591. 
Pyrometer, 9. 

Quadruple expansion engine, 461, 476, 550. 

R, 51, 52, 65, 66, 68, 70. 

Rankine, 151. 

Rankine cycle, 411, 424, 429, 447. 

Rankine's theorem, 106, 157, 158, 167. 

Rateau turbine, 524, 531, 538, 541. 

Rate of combustion, 560, 569. 

of evaporation, 569. 

of flame propagation, 309, 310, 319, 320, 
325. 
Ratio of expansion, 433, 436, 446, 447, 459. 

compound engines, 476. 

real and apparent, 450. 
Ratio of specific heats, 69, 70. 
Reaction turbine, 524, 533-535, 539. 
Real ratio of expansion, 450. 
Real specific heat, 61, 78. 
Reaumur thermometer, 8. 
Receiver compound engine, 464, 466-473. 
Receiver pressure, air compressor, 211. 
Recuperator, 281. 
Reevaporation, 431, 445, 448, 460. 
Reeves' method, 457. 

Refrigerating machine, 612, 616, 618-620, 
629, 633, 636, 637, 647, 658-660, 663, 
664. 
Refrigeration, 611-664. 

applications of, 649-657. 

compressed air, 227, 247. 

vapors used, 400-405. 
Regenerative expansion, 610, 612. 



Regenerator, 246, 257-259, 281, 541, 610. 
Regnault, 43, 46, 49. 
Regnault's law, 63. 
Regulation, air compressor, 238. 

gas engine, 336-338, 348, 349. 

steam engine, 462, 478. 
Reheating, 481. 
Reitlinger cycle, 253. 
Representation of heat absorbed, 106, 123, 

167. 
Reversibility, 139-141, 144-149. 

cycle, 138-141, 147, 148, 152, 175, 176. 

path, 125, 126, 162, 168, 175, 176. 
Rotary steam engine, 177, 192. 

Saturated steam, 356, 358-382. 

Saturated vapor, 356. 

Saturation curve, 365. 

Scales, thermometric, 8. 

Scavenging, 312, 327, 339. 

Second law of thermodynamics, 138-142, 

144-156. 
Siphon condenser, 584. 
Small calorie, 23. 
Soft coal, 560, 578. 
Solution, 15, 604. 
Sommeiller compressor, 240. 
Specific heat, 20, 21, 24-27, 57, 58. 

air, 71, 72. 

apparent, 61. 

difference, 65, 67, 77, 165. 

entropy diagram, 164. 

gases, 57-72. 

mean, 61, 164. 

negative, 115, 371. 

polytropics, 112, 115, 164. 

ratio, 68, 70. 

real, 61, 78. 

saturated vapor, 401. 

superheated steam, 383-385, 387, 388. 

volumetric, 60, 67. 

water, 24, 26, 359. 
Specific volume of steam, 360, 363, 368. 
Starting gas engines, 351. 
Steam, formation, 354-359, 366, 381. 

pressure-temperature relation, 355, 358, 
362, 368. 

saturated, 356, 358-382. 

superheated, 355, 358, 365, 366, 380, 
383-397. 
Steam adiabatic, 372, 373, 431, 432, 513, 

515, 517-520. 
Steam boiler, 566, 568-574. 
Steam consumption, 546, 553. 

from indicator diagram, 437, 487, 500, 
501. 
Steam cycles, 408-412, 414-418, 422-458, 

483, 514, 544. 
Steam engine, 419, 422-511, 550. 

cycle, 408-412, 414-418, 422-458, 483, 
514, 544. 

description, 422. 



INDEX 



437 



Steam engine, entropy diagram, 453-458, 
475. 

governing, 462, 478. 
Steam-ether engine, 483. 
Steam gas, 357, 390, 391. 
Steam jacket, 413, 438-441, 475, 482, 505. 
Steam loop, 583. 

Steam power plant, 407, 508, 560-590. 
Steam rate, 546, 553. 
Steam refrigeration, 631, 632, 638, 643. 
Steam table, 360, 367, 376, 405, 420, 421. 
Steam turbine, 512-542, 552, 553, 556. 
Steam, wet, 364, 367. 
Still, 591. 

Stirling hot-air engine, 260-268. 
Stoker, 578. 

Storage, compressed air, 185, 245. 
Straight-line compressor, 239. 
Stumpf turbine, 536. 
Sublimation, 17. 
Suction producer, 281, 282. 
Suction stroke, gas engine, 328. 
Sulphur dioxide, 404, 483, 606, 608, 611, 

631, 632, Table, p. 424. 
Superheat, locomotives, 443, 509, 554. 

refrigeration, 629, 633, 636, 647. 

turbines, 517, 552, 553. 

use of, 438, 442-444, 482, 551-553, 579. 
Superheated adiabatic, 416. 
Superheated steam, 355, 358, 365, 366, 380, 
383-397. 

cycles, 414-418. 

table, 421. 
Superheated vapor, 356. 
Superheaters, 579. 
Superheating calorimeter, 491. 
Surface condenser, 502, 584. 
Surface-condensing calorimeter, 490. 
Synopses, pp. 10, 17, 27-28, 37-38, 60, 69, 
76, 87-88, 125-127, 143-144, 193-195, 
249-252, 309-312, 346-347, 377-378, 
391-392, 420-421. 

Table, steam, 360, 367, 376, 405, 420, 421. 
Tandem-compound, 464, 466, 467, 469, 

471. 
Tank calorimeter, 489. 
Temperature, 6, 19-21. 

absolute, 152-156, 167. 

gas engine cylinder, 312, 314-318. 

inversion, 373, 395, 401. 

measurement, 6-9. 
Testing hot air engines, 274. 
Tests, locomotive, 497, 511, 554. 

refrigerating machine, 660-663. 

steam boiler, 572. 

steam engine, 484-505, 543-551, 553, 
555-559. 

steam turbine, 543-545, 552, 553, 555, 556. 
Thermal capacity, 57, 58. 
Thermal efficiency, 342. 
Thermal line, 83. 



Thermochemistry, 4, 40, 53, 56, 59. 
Thermodynamic function, 170. 
Thermodynamic surface, 84. 
Thermo-electric pyrometer, 9. 
Thermometer, 7, 8. 

air, 41, 42, 48, 49, 152. 
Thermometric scales, 8. 
Thermometry, 6-9. 
Theta-phi diagram, 170. 
Thomas' experiments, 385. 
Throttling, 388, 425, 426. 
Throttling calorimeter, 491. 
Throttling engine, 427, 507. 
Throttling, gas engine, 326, 348. 
Thrust in turbines, 528. 
Time of ignition, 321. 
Tonnage rating, 658, 659. 
Total heat-entropy diagram, 399. 
Total heat-pressure diagram, 399. 
Total heat, saturated steam, 359, 360, 388. 

superheated steam, 386. 
Tower, cooling, 585, 664. 
Transmission, air, 243-245. 

gas, 276. 
Transmissive efficiency, 212, 216, 243, 

244. 
Triple-expansion engine, 461, 476, 480, 549. 
Tubes in boilers, 569. 
Turbine, gas, 540. 

steam, 512-542, 552, 553, 556. 
Turbo-compressor, 239, 540. 
Turbo-locomotive, 540. 
Turbo-pump, 540. 

Two-cycle gas engine, 289-292, 329, 339. 
Two specific heats of gases, 57, 58, 62, 

64-72, 107, 165. 
Types, air compressor, 238-242. 

gas engine, 336-341, 350-351. 

locomotive, 509, 510. 

multiple-expansion engine, 461. 

steam engine, 507. 

vapor compressor, 664. 

Vacuum, footnote, p. 358. 
Vacuum distillation, 591-601. 
Vacuum pump, 236, 237, 584, 591. 
Valves, air compressor, 242. 

gas engine, 310, 326, 350. 

steam engine, 452, 468, 507. 
Vapor, paths, 392-399. 

specific heat, 401. 
Vapor adiabatic, 391-397. 
Vapor compressor, 624-638, 642, 658, 660, 

662, 664. 
Vapor refrigeration, 624-643, 647, 662. 
Vaporization, internal work, 359, 360. 

latent heat, 359, 360. 
Vaporizer, 279, 282, 310, 336, 612, 626. 
Vapors, 16, 17, 354-421. 

for refrigeration, 400-405, 630-632. 

saturated, 356. 

superheated, 356. 



438 



INDEX 



Velocity diagram, 527, 529, 534. 
Velocity turbine, 524, 530-533, 536-538. 
Velocity work, 127, 176, 512, 525, 518. 
Vibration work, 3, 12, 13, 54-56. 
Volume curves 377. 
Volumetric efficiency, 222-229. 
Volumetric specific heat, 66, 67. 
Volume, vapor, 360, 363, 368, 369, 401. 

Walls, gas engine cylinder, 312, 317, 325, 347. 

steam cylinder, 429, 431, 432, 504, 505. 

vapor compressor, 637. 
Warming machine, Kelvin, 623. 
Water, air compressor, 195-200, 206, 207. 

in refrigerating plant, 617, 635. 

specific heat, 24, 26, 359. 
Water gas, 278, 281, 329. 
Water jacket, air compressor, 201, 204 

gas engine, 312, 317, 325, 352, 353 
Water supply, evaporator, 600. 
Water-tube boiler, 568, 569. 
Watt's diagram, 86-90. 
Watt's law, 359. 
Waves, explosion, 319, 325. 



Westinghouse-Parsons turbine, 539. 

Wet compression, 629, 647. 

Wet steam, 364, 367. 

Willans' line, 556. 

Wiredrawing, 425, 426, 442, 448, 451 , 474, 518. 

Woolf engine, 463. 

Work, Clausius cycle, 410. 

compression, 210. 

external, 14, 15, 86-90, 95, 98, 121-123, 
160, 359, 374, 375. 

negative, 87, 89, 99. 

superheated adiabatic, 416. 

vapor adiabatic, 396. 

velocity, 127, 176, 512, 515, 518. 

vibration, 3, 12, 13, 54-56. 
Wormell's theorem, 36. 

y, 57, 58, 62, 64-72, 101, 102, 105, 107, 

110, 165. 
Yaryan evaporator, 595-600. 

Zero, absolute, 44, 45, 156. 

of entropy, 171. 
Zero line, 373. 



A LIST OF BOOKS 

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Steam and Steam Engineering. 



AUCHINCLOSS, W. S. The Practical Application of the Slide 
Valve and Link Motion to Stationary, Portable, Locomotive, 

and Marine Engines, with new and simple methods for propor- 
tioning the parts. Fifteenth Edition, revised. 52 Illustrations. 8vo. 
Cloth. 144 pp $2.00 

BACON, F. W. A Treatise on the Richards Steam Engine 
Indicator, with directions for its use. By Charles T. Porter. Re- 
vised, with notes and large additions as developed by American prac- 
tice; with an appendix containing useful formulae and rules for engi- 
neers. Fourth Edition. Illustrated. 16mo. Cloth. ISO pp 1.00 

BARRUS, G. H. Boiler Tests: Embracing the results of one 
hundred and thirty-seven evaporative tests, made on seventy-one boil- 
ers, conducted by the author. Illustrated. Svo. Cloth. 252 pp. 3. 00 

Engine Tests : Embracing the results of over one hundred feed- 
water tests and other investigations of various kinds of steam-engines, 
conducted by the author. With numerous figures, tables, and dia- 
grams. 8vo. Cloth. 338 pp.. 4.00 

The above two purchased together 6 . 00 

BEAUMONT, W. W. Practical Treatise on the Steam Engine 
Indicator, and Indicator Diagrams. With notes on engine per- 
formances, expansion of steam, behavior of steam in steam engine cylin- 
ders, and on gas- and oil-engine diagrams. Second Edition, revised and 
enlarged. 128 Illustrations. 8vo. Cloth. 261 pp Net, 2 . 50 

BEGTRUP, J. The Slide Valve and its Functions. With special refer- 
ence to modern practice in the United States. Second Edition. Illus- 
trated. 8vo. Cloth. 145 pp 2 .00 



2 D. VAN NOSTRAND COMPANY'S 

BERTIN, L. E. Marine Boilers: their Construction and Work- 
ing, dealing more especially with tubulous boilers. Translated by 
Leslie S. Robertson. Preface by Sir William White. Second Edition, 
revised and enlarged 350 Illustrations. 8vo. Cloth. 694pp. Net, 5.00 

BOOTH, W. H. Superheat, Superheating, and their Control. 
Illustrated. 8vo. Cloth. 170 pp Net, 1 .50 

— — Water Softening and Treatment, condensing plant, feed pumps, 
and heaters for steam users and manufacturers. Illustrated. 8vo. 
Cloth Net, 2.50 

CARPENTER, R. C, and DIEDERICHS, H. Internal Combus- 
tion Engines; Their Theory, Construction, and Operation. 
Illustrated. 8vo. Cloth. 611 pp Net, 5 .00 

CHRISTIE, W. W. Boiler-waters, Scale, Corrosion, Foaming. 
77 Illustrations. 8vo. Cloth. 242 pp Net, 3 . 00 

Chimney Design and Theory. A book for engineers and archi- 
tects. Revised and enlarged. Illustrated. 8vo. Cloth. 200 pp. 3. 00 

Furnace Draft ; its Production by Mechanical Methods. 

Second Edition, revised. Illustrated. 16mo. Boards. 80 pp. (Van 
Nostrand Science Series No. 123.) 50 cents 

CLARK, D. K. Fuel: its Combustion and Economy. 

Comprising an abridgment of "A Treatise on the Combustion of Coal," 
by C. W. Williams. With extensive additions on recent practice in the 
combustion and economy of fuel, coal, coke, wood, peat, petroleum, 
etc. Fourth Edition. 144 Illustrations. 12mo. Cloth. 366 pp.. 1.50 

DAY, C. Indicator Diagrams and Engine and Boiler Testing. 
Third Edition. 125 Illustrations. 12mo. Cloth. 220 pp 2.00 

DRAPER, C. H. Heat and the Principles of Thermodynamics. 
Third Edition. 133 Illustrations. 12mo. Cloth. 363 pp 1 . 50 

GOODEVE, T. M. A Text-book on the Steam Engine. With a 

supplement on gas engines. Twelfth Edition, enlarged. 143 Illustra- 
tions. 12mo Cloth 2 .00 

GOULD, E. S. The Arithmetic of the Steam Engine, illus- 
trated. 12mo. Cloth. 80 pp 1.00 

HAEDER, H. A Handbook on the Steam Engine. With 

especial reference to small and medium-sized engines. For the use 
of engine makers, mechanical draughtsmen, engineering students, and 
users of steam power. Translated from the German, with consid- 
erable additions and alterations, by H. H. P. Powles. Third English 
Edition, revised. Illustrated. 8vo. Cloth. 465 pp 3.00 



BOOKS ON STEAM AND ALLIED SUBJECTS. 3 

HALSEY, F. A. Slide Valve Gears. An explanation of the 
action and construction of plain and cut-off slide valves. Eleventh 
Edition, revised and, enlarged. 109 Illustrations. 12mo. Cloth. 211 
pp 1 . 50 

HAUSBRAND, E. Drying by Means of Air and Steam, with 
explanations, formulas, and tables, for use in practice. Translated 
from the German by A. C. Wright. Illustrated. 12mo Cloth. 
70 pp .....: Net, 2 .00 

— — Evaporating, Condensing, and Cooling Apparatus. Ex- 
planations, formula?, and tables for use in practice. Translated from 
the second revised German edition by A. C. Wright. With numerous 
figures, tables, and diagrams. 8vo. Cloth. 423 pp Net, 5.00 

HECK, R. C. H. The Steam Engine and Other Steam Motors, 
A text-book for engineering colleges and a treatise for engineers. In 
Two Volumes. 8vo, Cloth. 

Vol. I. The Thermodynamics and the Mechanics of the Engine. 

187 Illustrations. 400 pp Net, 3 . 50 

Vol. II. Form, Construction, and Working of the Engine : 

The Steam Turbine. 698 Illustrations. 686 pp Net, 5 .00 

Abridged edition of above two volumes In Press 

HUTTON, W. S. Steam Boiler Construction, a practical hand- 
book for engineers, boiler makers, and steam users, containing a large 
collection of rules and data relating to recent practice in the design, 
construction, and working of all kinds of stationary, locomotive, and 
marine steam boilers. Fourth Edition, carefidly revised and enlarged. 
540 Illustrations. 8vo. Cloth. 675 pp 6 .00 

The Practical Engineer's Handbook. Comprising a treatise on 

modern engines and boilers, marine, locomotive, and stationary. Sixth 
Edition, revised and enlarged. 423 Illustrations. 8vo. Cloth. 556 
pp 7 . 00 

JAMIESON, A. A Text-book on Steam and Steam Engines, 

including turbines and boilers. Specially arranged for the use of engi- 
neers qualifying for the Institution of Civil Engineers, the diplomas 
and degrees of technical colleges and universities, advanced science 
certificates of British and colonial Boards of Education, and honours 
certificates of the City and Guilds of London Institute, in mechanical 
engineering, and engineers generally. Fifteenth Edition, revised. 
Illustrated. 8vo. Cloth. 842 pp 3 .00 

Elementary Manual on Steam and the Steam Engine. 

Specially arranged for the use of first year science and art, City and 
Guilds of London Institute, and other elementary engineering students. 
Tenth Edition, revised. 12mo. Cloth 1 . 50 



4 D. VAN NOSTRAND COMPANY'S 

KENNEDY, R. Modern Engines and Power Generators. A 

practical work on prime movers and the transmission of power. With 
tables, figures, and full-page engravings. Six volumes. Illustrated. 

8yo. Cloth 15.00 

Single volumes, each 3.00 

KERSHAW, J. B. C. Fuel, Water, and Gas Analysis, 

for steam users. With 50 Illustrations. 8vo. Cloth. 177 pp. Net, 2.50 

KLEIN, J. F. Design of a High Speed Steam Engine. With 
notes, diagrams, formulas, and tables. Second Edition, revised and 
enlarged. 140 Illustrations. 8vo. Cloth. 257 pp Net, 5.00 

KLEINHANS, F. B. Boiler Construction. A practical explanation 
of the best modern methods of boiler construction, from the laying out 
of sheets to the completed boiler. With diagrams and full-page engrav- 
ings. 8vo. Cloth. 421 pp 3.00 

KOESTER, F. Steam Electric Power Plants and their Con- 
struction. A practical treatise on the design of central light and 
power stations and their economical construction and operation. 340 
Illustrations. 4to. Cloth. 473 pp Net, 5 .00 

LEASK, A. R. Triple and Quadruple Expansion Engines and 

Boilers and their Management. Fourth Edition, revised and 
enlarged. 74 Illustrations. 12mo. Cloth. 306 pp 2.00 

LEWES, V. B. Liquid and Gaseous Fuels and the Part They 
Play in Modern Power Production, illustrated. 8vo. Cloth. 
348 pp. (Van Nostrand's Westminster Series.) Net, 2.00 

LUCKE, C. E. Power, Cost, and Plant Designs and Construc- 
tion. 2 Vols In Press 

PICKWORTH, C. N. The Indicator Handbook. A practical man- 
ual for engineers. In Two Parts. 12mo. Cloth. 

Part I. The Indicator: Its Construction and Application. Third 
Edition. 81 Illustrations. 130 pp 1 . 50 

Part II. The Indicator Diagram: Its Analysis and Calculation. 
Third Edition. 148 Illustrations. 134 pp 1 . 50 

PRAY, T., Jr. Steam Tables and Engine Constant. Compiled 

from Regnault, Rankine, and Dixon directly, making use of the exact 
records. 8vo. Cloth. 126 pp 2 .00 

RANKINE, W. J. M. The Steam Engine and Other Prime 
Movers. With diagram of the mechanical properties of steam. Fold- 
ing-plates, numerous tables, and illustrations. Fifteenth Edition, thor- 
oughly revised by W. J. Millar. 8vo. Cloth 5.00 

■ Useful Rules and Tables Relating to Mensuration, Engi- 
neering, Structures, and Machines. With tables, tests, and 

formulae for the use of electrical engineers. By Andrew Jamieson. 
Seventh Edition, thoroughly revised by W. J. Millar. Illustrated. 12mo. 
Cloth. 482 pp 4.00 



BOOKS ON STEAM AND ALLIED SUBJECTS. 

RATEAU, A. Experimental Researches on the Flow of Steam 
Through Nozzles and Orifices. Translated by H. B. Brydon. 
Illustrated. 12mo. Cloth. 82 pp Net, 1 .50 

REED'S Marine Boilers. A treatise on the causes and prevention 
of their priming, with remarks on their general management. Third 
Edition, rewritten and enlarged. 79 Illustrations. 12mo. Cloth. 
264 pp Net, 2 .00 

RICHARDSON, J. The Modern Steam Engine. Theory, Design, 
Construction, Use. A practical treatise. 300 Illustrations. 8vo. 
Cloth. 384 pp Net, 3.50 

ROBERTSON, L. S. Water Tube Boilers. Based on a short course 
of lectures delivered at University College, London. With 171 Illus 
trations. 8vo. Cloth. 228 pp 3 .00 

ROSE, J. Key to Engines and Engine Running. A practical 
treatise upon the management of steam engines and boilers, for the 
use of those who desire to pass an examination to take charge of an 
engine or boiler. With numerous illustrations, and instructions upon 
engineers' calculations, indicator diagrams, engine adjustments, and 
other valuable information necessary for engineers and firemen. Third 
Edition. 12mo. Cloth. 416 pp 2 . 50 

ROSSITER, J. T. Steam Engines. Illustrated. 8vo. Cloth. (Van 
Nostrand's Westminster Series.) In Press 

ROWAN, F. J. The Practical Physics of the Modern Steam 
Boiler. With an introduction by R. H. Thurston. With 314 
Illustrations. 8vo. Cloth. 683 pp 7.50 

SCHUMANN, F. A Manual of Heating and Ventilation 

in their practical application, for the use of engineers and architects. 
Embracing tables and formulae for dimensions of heating, flow and 
return pipes for steam and hot water boilers, flues, etc. Fourth Edition, 
revised and enlarged. Illustrated. 12mo. Leather. 100 pp 1 .50 

SCRIBNER, J. M. Engineers' and Mechanics' Companion. 

Twenty-first Edition. Illustrated. 16mo. Morocco. 264 pp 1 .50 

SEATON, A. E. A Manual of Marine Engineering. Comprising 
the design, construction, and working of marine machinery. With 
numerous tables and illustrations reduced from working drawings. 
Sixteenth Edition, revised and enlarged. 8vo. Cloth. 735 pp 6.00 

SEATON, A. E., and ROUNTHWAITE, H. M. A Pocket-book of 
Marine Engineering Rules and Tables. For the use of 
marine engineers and naval architects, designers, draughtsmen, super- 
intendents, and . all engaged in the design and construction of marine 
machinery, naval and mercantile. Ninth Edition, revised and enlarged. 
Illustrated. 12mo. Leather. 560 pp 3 .00 



6 BOOKS ON STEAM AND ALLIED SUBJECTS. 

SEXTON, A. H. Fuel and Refractory Materials. Illustrated. 8vo. 
Cloth 2.00 

SHOCK, W. H. Steam Boilers : their design, construction, and man- 
agement. 150 Illustrations and 34 Full-page Plates. 4to. Half 
morocco. 475 pp 15 .00 

SOTHERN, J. W. The Marine Steam Turbine. A practical de- 
scription of the Parsons Marine Turbine as now constructed, fitted, 
and run. Second Edition, revised and enlarged. Illustrated. 8vo. 
Cloth. 173 pp Net, 2.50 

STILLMAN, P. Steam Engine Indicator and the improved manom- 
eter steam and vacuum gauges: their utility and apolication. New 
Edition. Illustrated. 16mo. Flexible cloth. 96 pp. . * 1 .00 

STODOLA, A. Steam Turbines. With an appendix on gas turbines, 
and the future of heat engines. Authorized translation by Louis C. 
Loewenstein. With 241 cuts and 3 lithographed tables. Second Revised 
Edition. 8vo. Cloth. 510 pp.. Net, 5.00 

TONGE, JAMES. Coal. Illustrated. 8vo. Cloth. 283 pp. (Van Nos- 
trand's Westminster Series.) Net, 2 . 00 

TRINKS, W., and HOUSUM, C. Shaft Governors. 27 Illustra- 
tions. 16mo. Boards. 100 pp. (Van Nostrand Science Series No. 
122.) 50 cents 

VAN NOSTRAND'S Year Book of Mechanical Engineering Data. 

With many tables and diagrams. (First year of issue, 1910.). . .In Press 

WALKER, SYDNEY F. Steam Boilers, Engines, and Turbines. 

189 Illustrations. 8vo. Cloth. 428 pp Net, 3 .00 

WATSON, E. P. Small Engines and Boilers. A manual of con- 
cise and specific directions for the construction of small steam engines 
and boilers of modern types from five horse-power down to model 
sizes. Illustrated. 12mo. Cloth. 116 pp 1 .25 

ZEUNER, A. Technical Thermodynamics. Translated from the 
Fifth, completely revised German Edition of Dr. Zeuner's original treat- 
ise on thermodynamics, by J. F. Klein. Illustrated. 8vo. Cloth. Two 
volumes Net, 8 . 00 

Any book in this list will be sent postpaid to any address in 
the world on receipt of price, by 

D. VAN NOSTRAND COMPANY 
Ipubliebers anb iBoofceellers 

23 Murray and 27 Warren Streets, New York. 



SEP 08 1910 



One copy del. to Cat. Div. 



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